eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 6 Issue 1 - 2017
Rani Distribution and Its Application
Rama Shanker*
Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea
Received: May 19, 2017 | Published: May 30, 2017
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R (2017) Rani Distribution and Its Application. Biom Biostat Int J 6(1): 00155. DOI: 10.15406/bbij.2017.06.00155

Abstract

In the present paper, a new one parameter lifetime distribution named, “Rani Distribution’ has been proposed for modeling lifetime data from engineering and biomedical sciences. Its various statistical and mathematical properties including its shapes for varying values of parameter, moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, deviations from the mean and the median, Bonferroni and Lorenz curves, order statistics , Renyi entropy measure and stress-strength reliability have been studied. Both the maximum likelihood estimation and the method of moments have been discussed for estimating the parameter of the proposed distribution. A simulation study has been carried out and results are presented. A numerical example has been presented to test the goodness of fit of the proposed distribution and it has been found that it gives much closer fit than almost all one parameter lifetime distributions introduced in statistical literature.

Keywords: Lifetime Distributions; Statistical and Mathematical Properties; Parameter Estimation; Goodness of Fit

Introducton

In the present world the modeling and analyzing lifetime data are essential in almost all applied sciences including medicine, engineering, insurance and finance, amongst others. The two classical one parameter lifetime distributions which are popular and are in use for modeling lifetime data from biomedical science and engineering are exponential and Lindley introduced by Lindley [1]. Shanker, et al. [2] have detailed comparative study on modeling of lifetime data from various fields of knowledge and observed that there are many lifetime data where these two distributions are not suitable due to their shapes, nature of hazard rate functions, and mean residual life functions, amongst others. In search for new one parameter lifetime distributions which gives better fit than exponential and Lindley distributions, recently Shanker has introduced several one parameter lifetime distributions in statistical literature namely Akash [3], Shanker [4], Aradhana [5], Sujatha [6], Amarendra [7], Devya [8], Rama [9] and Akshaya [10] and showed that these distributions gives better fit than the classical exponential and Lindley distributions. The probability density function (pdf) and the corresponding cumulative distribution function (cdf) of Akash[3], Shanker [4], Aradhana [5], Sujatha [6], Amarendra [7], Devya [8], Rama [9] and Lindley [1] distributions are presented in Table (1). It has also been discussed by Shanker that although each of these lifetime distributions has advantages and disadvantages over one another due to its shapes, hazard rate functions and mean residual life functions, there are still many lifetime data where these distributions are not suitable for modeling lifetime data from theoretical or applied point of view. Therefore, an attempt has been made in this paper to obtain a new lifetime distribution which is flexible than these one parameter lifetime distributions for modeling lifetime data in reliability and in terms of its hazard rate shapes.

 The new one parameter lifetime distribution is based on a two-component mixture of an exponential distribution having scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and a gamma distribution having shape parameter 5 and scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  with their mixing proportion θ 5 θ 5 +24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeqiUde3aaWbaaeqajyaGbaGaaGynaaaaaKqbagaacqaH4oqCdaah aaqabKGbagaacaaI1aaaaKqbakabgUcaRiaaikdacaaI0aaaaaaa@40AB@ .

The probability density function (p.d.f.) of a new one parameter lifetime distribution can be introduced as

f( x;θ )= θ 5 θ 5 +24 ( θ+ x 4 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadIhacaGG7aGaeqiUdehacaGLOaGaayzkaaGaeyypa0Za aSaaaeaacqaH4oqCdaahaaqabKGbagaacaaI1aaaaaqcfayaaiabeI 7aXnaaCaaabeqcgayaaiaaiwdaaaqcfaOaey4kaSIaaGOmaiaaisda aaWaaeWaaeaacqaH4oqCcqGHRaWkcaWG4bWaaWbaaeqajyaGbaGaaG inaaaaaKqbakaawIcacaGLPaaacaWGLbWaaWbaaeqajyaGbaGaeyOe I0IaeqiUdeNaamiEaaaajuaGcaaMc8UaaGPaVlaaykW7caaMc8Uaai 4oaiaadIhacqGH+aGpcaaIWaGaaiilaiaaykW7caaMc8UaeqiUdeNa eyOpa4JaaGimaaaa@6563@  (1.1)

 We would call this distribution, “Rani distribution”. This distribution can be easily expressed as a mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@  and gamma ( 5,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaI1aGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B32@  with mixing proportion θ 5 θ 5 +24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeqiUde3aaWbaaeqajyaGbaGaaGynaaaaaKqbagaacqaH4oqCjyaG daahaaqabeaacaaI1aaaaKqbakabgUcaRiaaikdacaaI0aaaaaaa@40AB@ . We have

f( x,θ )=p g 1 ( x )+( 1p ) g 2 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacYcacqaH4oqCaiaawIcacaGLPaaacqGH9aqp caWGWbGaaGPaVlaadEgalmaaBaaajyaGbaqcLbmacaaIXaaajyaGbe aajuaGdaqadaqaaiaadIhaaiaawIcacaGLPaaacqGHRaWkdaqadaqa aiaaigdacqGHsislcaWGWbaacaGLOaGaayzkaaGaam4zamaaBaaaju aibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGaayzkaaaa aa@5127@

where p= θ 5 θ 5 +24 , g 1 ( x )=θ e θx ,and g 2 ( x )= θ 5 x 4 e θx 24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadchacq GH9aqpdaWcaaqaaiabeI7aXLGbaoaaCaaabeqaaiaaiwdaaaaajuaG baGaeqiUde3aaWbaaeqajyaGbaGaaGynaaaajuaGcqGHRaWkcaaIYa GaaGinaaaacaGGSaGaaGPaVlaaykW7caWGNbWaaSbaaeaacaaIXaaa beaadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcqaH4oqCca WGLbqcga4aaWbaaeqabaGaeyOeI0IaeqiUdeNaaGPaVlaadIhaaaqc faOaaiilaiaaykW7caaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlaadE gajyaGdaWgaaqaaiaaikdaaeqaaKqbaoaabmaabaGaamiEaaGaayjk aiaawMcaaiabg2da9maalaaabaGaeqiUdexcga4aaWbaaeqabaGaaG ynaaaajuaGcaWG4bWaaWbaaeqajyaGbaGaaGinaaaajuaGcaWGLbWa aWbaaeqajyaGbaGaeyOeI0IaeqiUdeNaamiEaaaaaKqbagaacaaIYa GaaGinaaaaaaa@714A@ .

The corresponding cumulative distribution function (c.d.f.) of (1.1) can easily be obtained as

F( x,θ )=1[ 1+ θx( θ 3 x 3 +4 θ 2 x 2 +12θx+24 ) θ 5 +24 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacYcacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaaGymaiabgkHiTKqbaoaadmaakeaajugibi aaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacqaH4oqCcaWG4bqcfa4a aeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaio daaaqcLbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqc LbsacqGHRaWkcaaI0aGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaikdaaaqcLbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaikda aaqcLbsacqGHRaWkcaaIXaGaaGOmaiabeI7aXjaadIhacqGHRaWkca aIYaGaaGinaaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaGdaah aaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaikdacaaI0a aaaaGccaGLBbGaayzxaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaa jugWaiabgkHiTiabeI7aXjaaykW7caWG4baaaKqzGeGaaGPaVlaayk W7caGG7aGaamiEaiabg6da+iaaicdacaGGSaGaeqiUdeNaeyOpa4Ja aGimaaaa@83FA@  (1.2)

The graphs of the p.d.f. and the c.d.f. of Rani distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@ are shown in Figures 1 & 2. The p.d.f. of Rani distribution is monotonically decreasing.

Distributions

Probability density functions and Cumulative distribution functions

Akash

pdf

f( x )= θ 3 θ 2 +2 ( 1+ x 2 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcgayaaiaaiodaaaaajuaGbaGaeqiUde3aaWbaaeqajy aGbaGaaGOmaaaajuaGcqGHRaWkcaaIYaaaamaabmaabaGaaGymaiab gUcaRiaadIhadaahaaqabKGbagaacaaIYaaaaaqcfaOaayjkaiaawM caaiaadwgadaahaaqabKGbagaacqGHsislcqaH4oqCcaaMc8UaamiE aaaajuaGcaaMc8UaaGPaVlaacUdacaWG4bGaeyOpa4JaaGimaiaacY cacaaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@5E18@

cdf

F( x )=1[ 1+ θx( θx+2 ) θ 2 +2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaaykW7caWG4b WaaeWaaeaacqaH4oqCcaaMc8UaamiEaiabgUcaRiaaikdaaiaawIca caGLPaaaaeaacqaH4oqCdaahaaqabeaajugWaiaaikdaaaqcfaOaey 4kaSIaaGOmaaaaaiaawUfacaGLDbaacaWGLbWcdaahaaqcgayabeaa jugWaiabgkHiTiabeI7aXjaaykW7caWG4baaaiaaykW7caaMc8oaaa@5CC8@

Shanker

pdf

f( x )= θ 2 θ 2 +1 ( θ+x ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiaaykW7caaMc8Uaeyypa0Ja aGPaVlaaykW7daWcaaqaaiabeI7aXnaaCaaajuaibeqaaiaaikdaaa aajuaGbaGaeqiUde3aaWbaaKqbGeqabaGaaGOmaaaajuaGcqGHRaWk caaIXaaaaiaaykW7caaMc8+aaeWaaeaacqaH4oqCcqGHRaWkcaWG4b aacaGLOaGaayzkaaGaaGPaVlaaykW7caWGLbWaaWbaaKqbGeqabaGa eyOeI0IaeqiUdeNaamiEaaaajugWaiaaykW7aaa@5C51@

cdf

F( x )=1[ 1+ θx θ 2 +1 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacq aH4oqCdaahaaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaaGymaaaa aiaawUfacaGLDbaacaaMe8UaamyzamaaCaaajuaibeqaaiabgkHiTi abeI7aXjaadIhaaaqcfaOaaGPaVdaa@5142@

Aradhana

pdf

f( x )= θ 3 θ 2 +2θ+2 ( 1+x ) 2 e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3cdaahaaqcfayabeaajugWaiaaiodaaaaajuaGbaGaeqiUde3aaW baaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGH RaWkcaaIYaaaamaabmaabaGaaGymaiabgUcaRiaadIhaaiaawIcaca GLPaaadaahaaqabeaajugWaiaaikdaaaqcfaOaamyzamaaCaaabeqa aKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@55CE@

cdf

F( x )=1[ 1+ θx( θx+2θ+2 ) θ 2 +2θ+2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhadaqada qaaiabeI7aXjaadIhacqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOm aaGaayjkaiaawMcaaaqaaiabeI7aXnaaCaaabeqaaKqzadGaaGOmaa aajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOmaaaaaiaawUfa caGLDbaacaWGLbWcdaahaaqcgayabeaajugWaiabgkHiTiabeI7aXj aadIhaaaGaaGPaVdaa@5D44@

Sujatha

pdf

f( x )= θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaIZaaaaaqcfayaaiabeI7aXTWaaWbaaK qbagqabaqcLbmacaaIYaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaa ikdaaaWaaeWaaeaacaaIXaGaey4kaSIaamiEaiabgUcaRiaadIhada ahaaqabeaajugWaiaaikdaaaaajuaGcaGLOaGaayzkaaGaamyzaSWa aWbaaKqbagqabaqcLbmacqGHsislcqaH4oqCcaWG4baaaaaa@578A@

cdf

F( x )=1[ 1+ θx( θx+θ+2 ) θ 2 +θ+2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhadaqada qaaiabeI7aXjaadIhacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaacaGL OaGaayzkaaaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIYaaaaKqbak abgUcaRiabeI7aXjabgUcaRiaaikdaaaaacaGLBbGaayzxaaGaamyz aSWaaWbaaKqbagqabaqcLbmacqGHsislcqaH4oqCcaWG4baaaaaa@5A40@

Amarendra

pdf

f( x )= θ 4 θ 3 + θ 2 +2θ+6 ( 1+x+ x 2 + x 3 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaI0aaaaaqcfayaaiabeI7aXnaaCaaabe qaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCdaahaaqabeaajugW aiaaikdaaaqcfaOaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaiAdaaa WaaeWaaeaacaaIXaGaey4kaSIaamiEaiabgUcaRiaadIhadaahaaqa beaajugWaiaaikdaaaqcfaOaey4kaSIaamiEaSWaaWbaaKqbagqaba qcLbmacaaIZaaaaaqcfaOaayjkaiaawMcaaiaadwgalmaaCaaajuaG beqaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@61F8@

cdf

F( x )=1[ 1+ θ 3 x 3 + θ 2 ( θ+3 ) x 2 +θ( θ 2 +2θ+6 )x θ 3 + θ 2 +2θ+6 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeeaY=PjY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGgbWaae WaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaGymaiabgkHiTmaa dmaabaGaaGymaiabgUcaRmaalaaabaGaeqiUde3aaWbaaeqabaqcLb macaaIZaaaaKqbakaadIhajyaGdaahaaqabeaacaaIZaaaaKqbakab gUcaRiabeI7aXLGbaoaaCaaabeqaaiaaikdaaaqcfa4aaeWaaeaacq aH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaGaamiEaKGbaoaaCaaa beqaaiaaikdaaaqcfaOaey4kaSIaeqiUde3aaeWaaeaacqaH4oqCjy aGdaahaaqabeaacaaIYaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGH RaWkcaaI2aaacaGLOaGaayzkaaGaamiEaaqaaiabeI7aXnaaCaaabe qcgayaaiaaiodaaaqcfaOaey4kaSIaeqiUde3aaWbaaeqajyaGbaGa aGOmaaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOnaaaaai aawUfacaGLDbaacaWGLbqcga4aaWbaaeqabaGaeyOeI0IaeqiUdeNa amiEaaaaaaa@7559@

Devya

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f( x )= θ 5 θ 4 + θ 3 +2 θ 2 +6θ+24 ( 1+x+ x 2 + x 3 + x 4 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaI1aaaaaqcfayaaiabeI7aXnaaCaaabe qaaKqzadGaaGinaaaajuaGcqGHRaWkcqaH4oqCdaahaaqabeaajugW aiaaiodaaaqcfaOaey4kaSIaaGOmaiabeI7aXnaaCaaabeqaaKqzad GaaGOmaaaajuaGcqGHRaWkcaaI2aGaeqiUdeNaey4kaSIaaGOmaiaa isdaaaWaaeWaaeaacaaIXaGaey4kaSIaamiEaiabgUcaRiaadIhalm aaCaaajuaGbeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaWG4bWaaWba aeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaadIhadaahaaqabeaaju gWaiaaisdaaaaajuaGcaGLOaGaayzkaaGaamyzamaaCaaabeqaaKqz adGaeyOeI0IaeqiUdeNaamiEaaaaaaa@6C89@

cdf

F( x )=1[ 1+ { θ 4 ( x 4 + x 3 + x 2 +x )+ θ 3 ( 4 x 3 +3 x 2 +2x ) +6 θ 2 ( 2 x 2 +x )+24θx } θ 4 + θ 3 +2 θ 2 +6θ+24 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaamaacmaaeaqabeaacqaH4o qClmaaCaaajuaGbeqaaKqzadGaaGinaaaajuaGdaqadaqaaiaadIha lmaaCaaajuaGbeqaaKqzadGaaGinaaaajuaGcqGHRaWkcaWG4bWaaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaadIhadaahaaqabeaa jugWaiaaikdaaaqcfaOaey4kaSIaamiEaaGaayjkaiaawMcaaiabgU caRiabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGdaqadaqaaiaa isdacaWG4bWaaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaio dacaWG4bWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaikda caWG4baacaGLOaGaayzkaaaabaGaey4kaSIaaGOnaiabeI7aXnaaCa aabeqaaKqzadGaaGOmaaaajuaGdaqadaqaaiaaikdacaWG4bWcdaah aaqcfayabeaajugWaiaaikdaaaqcfaOaey4kaSIaamiEaaGaayjkai aawMcaaiabgUcaRiaaikdacaaI0aGaeqiUdeNaamiEaaaacaGL7bGa ayzFaaaabaGaeqiUde3aaWbaaeqabaqcLbmacaaI0aaaaKqbakabgU caRiabeI7aXTWaaWbaaKqbagqabaqcLbmacaaIZaaaaKqbakabgUca RiaaikdacqaH4oqClmaaCaaajuaGbeqaaKqzadGaaGOmaaaajuaGcq GHRaWkcaaI2aGaeqiUdeNaey4kaSIaaGOmaiaaisdaaaaacaGLBbGa ayzxaaGaamyzaSWaaWbaaKqbagqabaqcLbmacqGHsislcqaH4oqCca WG4baaaaaa@9AD6@

Rama

pdf

f( x )= θ 4 θ 3 +6 ( 1+ x 3 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaI0aaaaaqcfayaaiabeI7aXTWaaWbaaK qbagqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaiAdaaaWaaeWaaeaa caaIXaGaey4kaSIaamiEaSWaaWbaaKqbagqabaqcLbmacaaIZaaaaa qcfaOaayjkaiaawMcaaiaadwgadaahaaqabeaajugWaiabgkHiTiab eI7aXjaadIhaaaaaaa@531A@

cdf

F( x )=1[ 1+ θ 3 x 3 +3 θ 2 x 2 +6θx θ 3 +6 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXTWaaWbaaKqbag qabaqcLbmacaaIZaaaaKqbakaadIhadaahaaqabeaajugWaiaaioda aaqcfaOaey4kaSIaaG4maiabeI7aXnaaCaaabeqaaKqzadGaaGOmaa aajuaGcaWG4bWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaa iAdacqaH4oqCcaaMc8UaamiEaaqaaiabeI7aXnaaCaaabeqaaKqzad GaaG4maaaajuaGcqGHRaWkcaaI2aaaaaGaay5waiaaw2faaiaadwga lmaaCaaajuaGbeqaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@6470@

Akshaya

pdf

f( x )= θ 4 θ 3 +3 θ 2 +6θ+6 ( 1+x ) 3 e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3cdaahaaqcfayabeaajugWaiaaisdaaaaajuaGbaGaeqiUde3aaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaiodacqaH4oqCdaah aaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaaGOnaiabeI7aXjabgU caRiaaiAdaaaWaaeWaaeaacaaIXaGaey4kaSIaamiEaaGaayjkaiaa wMcaamaaCaaabeqaaKqzadGaaG4maaaajuaGcaWGLbWaaWbaaeqaba qcLbmacqGHsislcqaH4oqCcaaMc8UaamiEaaaaaaa@5D53@

cdf

F( x )=1[ 1+ θ 3 x 3 +3 θ 2 ( θ+1 ) x 2 +3θ( θ 2 +2θ+2 )x θ 3 +3 θ 2 +6θ+6 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXTWaaWbaaKqbag qabaqcLbmacaaIZaaaaKqbakaadIhalmaaCaaajuaGbeqaaKqzadGa aG4maaaajuaGcqGHRaWkcaaIZaGaeqiUde3cdaahaaqcfayabeaaju gWaiaaikdaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGL OaGaayzkaaGaamiEamaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRa WkcaaIZaGaeqiUde3aaeWaaeaacqaH4oqCdaahaaqabeaajugWaiaa ikdaaaqcfaOaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaikdaaiaawI cacaGLPaaacaWG4baabaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaa aKqbakabgUcaRiaaiodacqaH4oqClmaaCaaajuaGbeqaaKqzadGaaG OmaaaajuaGcqGHRaWkcaaI2aGaeqiUdeNaey4kaSIaaGOnaaaaaiaa wUfacaGLDbaacaWGLbWaaWbaaeqabaqcLbmacqGHsislcqaH4oqCca aMc8UaamiEaaaaaaa@7E8D@

Lindley

pdf

f( x )= θ 2 θ+1 ( 1+x ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaIYaaaaaqcfayaaiabeI7aXjabgUcaRi aaigdaaaWaaeWaaeaacaaIXaGaey4kaSIaamiEaaGaayjkaiaawMca aiaadwgadaahaaqabeaajugWaiabgkHiTiabeI7aXjaaykW7caWG4b aaaaaa@4E36@

cdf

F( x )=1[ 1+ θx θ+1 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaaykW7caWG4b aabaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaacaWGLbWa aWbaaeqabaqcLbmacqGHsislcqaH4oqCcaaMc8UaamiEaaaaaaa@4F18@

Table 1: pdf and cdf of Akash, Shanker, Aradhana, Sujatha, Amarendra, Devya, Rama, Akshaya and Lindley distributions for x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abg6da+iaaicdacaGGSaGaaGPaVlabeI7aXjabg6da+iaaicdaaaa@3EF6@ .

Figure 1: Graphs of the pdf of Rani distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Figure 2: Graphs of the cdf of Rani distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Moments and Moments Based Measures

The moment generating function of Rani distribution (1.1) can be obtained as

M X ( t )= θ 5 θ 5 +24 0 e ( θt )x ( θ+ x 4 ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eada WgaaqcgayaaiaadIfaaKqbagqaamaabmaabaGaamiDaaGaayjkaiaa wMcaaiabg2da9maalaaabaGaeqiUde3aaWbaaeqajyaGbaGaaGynaa aaaKqbagaacqaH4oqCdaahaaqabKGbagaacaaI1aaaaKqbakabgUca RiaaikdacaaI0aaaamaapehabaGaamyzaKGbaoaaCaaabeqaaiabgk HiTmaabmaabaGaeqiUdeNaeyOeI0IaamiDaaGaayjkaiaawMcaaiaa ykW7caWG4baaaKqbaoaabmaabaGaeqiUdeNaey4kaSIaamiEamaaCa aabeqcgayaaiaaisdaaaaajuaGcaGLOaGaayzkaaGaaGPaVdqcgaya aiaaicdaaeaacqGHEisPaKqbakabgUIiYdGaamizaiaadIhaaaa@6208@
= θ 5 θ 5 +24 [ θ θt + 24 ( θt ) 5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaGynaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaaajuaGdaWadaGc baqcfa4aaSaaaOqaaKqzGeGaeqiUdehakeaajugibiabeI7aXjabgk HiTiaadshaaaGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGOmaiaaisda aOqaaKqbaoaabmaakeaajugibiabeI7aXjabgkHiTiaadshaaOGaay jkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaaaaaOGa ay5waiaaw2faaaaa@5BD1@
= θ 5 θ 5 +24 [ k=0 ( t θ ) k + 24 θ 5 k=0 ( k+4 k ) ( t θ ) k ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaGynaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaaajuaGdaWadaGc baqcfa4aaabCaOqaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsaca WG0baakeaajugibiabeI7aXbaaaOGaayjkaiaawMcaaKqbaoaaCaaa leqajeaibaqcLbmacaWGRbaaaKqzGeGaey4kaScaleaajugibiaadU gacqGH9aqpcaaIWaaaleaajugibiabg6HiLcGaeyyeIuoajuaGdaWc aaGcbaqcLbsacaaIYaGaaGinaaGcbaqcLbsacqaH4oqCjuaGdaahaa WcbeqcbasaaKqzadGaaGynaaaaaaqcfa4aaabCaOqaaKqbaoaabmaa keaajugibuaabeqaceaaaOqaaKqzGeGaam4AaiabgUcaRiaaisdaaO qaaKqzGeGaam4AaaaaaOGaayjkaiaawMcaaKqbaoaabmaakeaajuaG daWcaaGcbaqcLbsacaWG0baakeaajugibiabeI7aXbaaaOGaayjkai aawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbaaaaWcbaqcLbsa caWGRbGaeyypa0JaaGimaaWcbaqcLbsacqGHEisPaiabggHiLdaaki aawUfacaGLDbaaaaa@7C2E@
= k=0 θ 5 +( k+1 )( k+2 )( k+3 )( k+4 ) θ 5 +24 ( t θ ) k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWba aSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGHRaWkjuaGdaqadaGcba qcLbsacaWGRbGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaeWa aOqaaKqzGeGaam4AaiabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbao aabmaakeaajugibiaadUgacqGHRaWkcaaIZaaakiaawIcacaGLPaaa juaGdaqadaGcbaqcLbsacaWGRbGaey4kaSIaaGinaaGccaGLOaGaay zkaaaabaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGyn aaaajugibiabgUcaRiaaikdacaaI0aaaaaWcbaqcLbsacaWGRbGaey ypa0JaaGimaaWcbaqcLbsacqGHEisPaiabggHiLdqcfa4aaeWaaOqa aKqbaoaalaaakeaajugibiaadshaaOqaaKqzGeGaeqiUdehaaaGcca GLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaadUgaaaaaaa@6DA3@
Thus the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th moment about origin μ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadkhaaeqaaOWaaWbaaSqabeaakiadacUHYaIOaaaaaa@3BF8@ , obtained as the coefficient of t r r! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamiDaKqbaoaaCaaaleqajeaibaqcLbmacaWGYbaaaaGc baqcLbsacaWGYbGaaiyiaaaaaaa@3D65@  in M X ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaKqaGeaajugWaiaadIfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiDaaGccaGLOaGaayzkaaaaaa@3DF9@ , of Rani distribution can be given by

μ r = r![ θ 5 +( r+1 )( r+2 )( r+3 )( r+4 ) ] θ r ( θ 5 +24 ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcKfaG=haajugWaiaadkhaaSqabaqcfa4aaWbaaSqa bKqaGeaajugibiadacUHYaIOaaGaeyypa0tcfa4aaSaaaOqaaKqzGe GaamOCaiaacgcajuaGdaWadaGcbaqcLbsacqaH4oqCjuaGdaahaaWc beqcKfaG=haajugWaiaaiwdaaaqcLbsacqGHRaWkjuaGdaqadaGcba qcLbsacaWGYbGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaeWa aOqaaKqzGeGaamOCaiabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbao aabmaakeaajugibiaadkhacqGHRaWkcaaIZaaakiaawIcacaGLPaaa juaGdaqadaGcbaqcLbsacaWGYbGaey4kaSIaaGinaaGccaGLOaGaay zkaaaacaGLBbGaayzxaaaabaqcLbsacqaH4oqCjuaGdaahaaWcbeqc KfaG=haajugWaiaadkhaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa 4aaWbaaSqabKazba4=baqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOm aiaaisdaaOGaayjkaiaawMcaaaaajugibiaaykW7caaMc8UaaGPaVl aacUdacaWGYbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaioda caGGSaGaaiOlaiaac6cacaGGUaaaaa@8697@  (2.1)

Substituting r=1,2,3,and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaaykW7caaM c8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7caaMc8UaaGPaVlaais daaaa@48F3@ , the first four moments about origin of Rani distribution are obtained as
μ 1 = θ 5 +120 θ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaahaaWcbeqa aKqzGeGamai4gkdiIcaacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaH4o qCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaa igdacaaIYaGaaGimaaGcbaqcLbsacqaH4oqCjuaGdaqadaGcbaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiab gUcaRiaaikdacaaI0aaakiaawIcacaGLPaaaaaaaaa@561D@ , μ 2 = 2( θ 5 +360 ) θ 2 ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaahaaWcbeqa aKqzGeGamai4gkdiIcaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIYa qcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaiwdaaaqcLbsacqGHRaWkcaaIZaGaaGOnaiaaicdaaOGaayjkai aawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaa ikdaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGe aajugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaGccaGLOaGa ayzkaaaaaaaa@5C5F@ , μ 3 = 6( θ 5 +840 ) θ 3 ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaG4maaWcbeaajuaGdaahaaWcbeqa aKqzGeGamai4gkdiIcaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaI2a qcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaiwdaaaqcLbsacqGHRaWkcaaI4aGaaGinaiaaicdaaOGaayjkai aawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaa iodaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGe aajugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaGccaGLOaGa ayzkaaaaaaaa@5C68@ , μ 4 = 24( θ 5 +1680 ) θ 4 ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGinaaWcbeaajuaGdaahaaWcbeqa aKqzGeGamai4gkdiIcaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIYa GaaGinaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGymaiaaiAdacaaI4aGaaG imaaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaGdaahaaWcbeqc basaaKqzadGaaGinaaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGda ahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaikdacaaI 0aaakiaawIcacaGLPaaaaaaaaa@5DE1@
Now using relationship between central moments and moments about origin, the central moments of Rani distribution are obtained as

μ 2 = θ 10 +528 θ 5 +2880 θ 2 ( θ 5 +24 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmaca aIXaGaaGimaaaajugibiabgUcaRiaaiwdacaaIYaGaaGioaiabeI7a XLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaG OmaiaaiIdacaaI4aGaaGimaaGcbaqcLbsacqaH4oqCjuaGdaahaaWc beqcbasaaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaikda caaI0aaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaG Omaaaaaaaaaa@61CE@
μ 3 = 2( θ 15 +1512 θ 10 +1728 θ 5 +69120 ) θ 3 ( θ 5 +24 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaG4maaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiaaikdajuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGymaiaaiwdaaaqcLbsacqGHRaWk caaIXaGaaGynaiaaigdacaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGe aajugWaiaaigdacaaIWaaaaKqzGeGaey4kaSIaaGymaiaaiEdacaaI YaGaaGioaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaK qzGeGaey4kaSIaaGOnaiaaiMdacaaIXaGaaGOmaiaaicdaaOGaayjk aiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaiodaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqa GeaajugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaGccaGLOa Gaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaaaaaaa@7059@
μ 4 = 9( θ 20 +2656 θ 15 +58752 θ 10 +1234944 θ 5 +3870720 ) θ 4 ( θ 5 +24 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGinaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiaaiMdajuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGOmaiaaicdaaaqcLbsacqGHRaWk caaIYaGaaGOnaiaaiwdacaaI2aGaeqiUdexcfa4aaWbaaSqabKqaGe aajugWaiaaigdacaaI1aaaaKqzGeGaey4kaSIaaGynaiaaiIdacaaI 3aGaaGynaiaaikdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG ymaiaaicdaaaqcLbsacqGHRaWkcaaIXaGaaGOmaiaaiodacaaI0aGa aGyoaiaaisdacaaI0aGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaiwdaaaqcLbsacqGHRaWkcaaIZaGaaGioaiaaiEdacaaIWaGaaG4n aiaaikdacaaIWaaakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaaI0aaaaKqbaoaabmaakeaajugibiab eI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaS IaaGOmaiaaisdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqc LbmacaaI0aaaaaaaaaa@7E8E@

The coefficient of variation ( C.V ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaam4qaiaac6cacaWGwbaakiaawIcacaGLPaaaaaa@3B05@ , coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaakaaakeaajugibiabek7aILqbaoaaBaaajeaibaqcLbma caaIXaaaleqaaaqabaaakiaawIcacaGLPaaaaaa@3DC6@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaa kiaawIcacaGLPaaaaaa@3D1F@ and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeq4SdCgakiaawIcacaGLPaaaaaa@3A57@  of Rani distribution are thus obtained as

C.V= σ μ 1 = θ 10 +528 θ 5 +2880 θ 5 +120 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb GaaiOlaiaadAfacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCaOqa aKqzGeGaeqiVd0wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcfa 4aaWbaaSqabeaajugibiadacUHYaIOaaaaaiabg2da9Kqbaoaalaaa keaajuaGdaGcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGymaiaaicdaaaqcLbsacqGHRaWkcaaI1aGaaGOmaiaaiIda cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgU caRiaaikdacaaI4aGaaGioaiaaicdaaSqabaaakeaajugibiabeI7a XLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaG ymaiaaikdacaaIWaaaaaaa@63BA@
β 1 = μ 3 μ 2 3/2 = 2( θ 15 +1512 θ 10 +1728 θ 5 +69120 ) ( θ 10 +528 θ 5 +2880 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaa beaajugibiabg2da9KqbaoaalaaakeaajugibiabeY7aTLqbaoaaBa aajeaibaqcLbmacaaIZaaaleqaaaGcbaqcLbsacqaH8oqBjuaGdaWg aaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaahaaWcbeqcbasaaKqzad GaaG4maiaac+cacaaIYaaaaaaajugibiabg2da9Kqbaoaalaaakeaa jugibiaaikdajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbe qcbasaaKqzadGaaGymaiaaiwdaaaqcLbsacqGHRaWkcaaIXaGaaGyn aiaaigdacaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaig dacaaIWaaaaKqzGeGaey4kaSIaaGymaiaaiEdacaaIYaGaaGioaiab eI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaS IaaGOnaiaaiMdacaaIXaGaaGOmaiaaicdaaOGaayjkaiaawMcaaaqa aKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLb macaaIXaGaaGimaaaajugibiabgUcaRiaaiwdacaaIYaGaaGioaiab eI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaS IaaGOmaiaaiIdacaaI4aGaaGimaaGccaGLOaGaayzkaaqcfa4aaWba aSqabKqaGeaajugWaiaaiodacaGGVaGaaGOmaaaaaaaaaa@885D@
β 2 = μ 4 μ 2 2 = 9( θ 20 +2656 θ 15 +58752 θ 10 +1234944 θ 5 +3870720 ) ( θ 10 +528 θ 5 +2880 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiabeY7aTLqbaoaaBaaajeaibaqcLbmacaaI0a aaleqaaaGcbaqcLbsacqaH8oqBjuaGdaWgaaqcbasaaKqzadGaaGOm aaWcbeaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaaqcLbsacq GH9aqpjuaGdaWcaaGcbaqcLbsacaaI5aqcfa4aaeWaaOqaaKqzGeGa eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdacaaIWaaaaKqzGe Gaey4kaSIaaGOmaiaaiAdacaaI1aGaaGOnaiabeI7aXLqbaoaaCaaa leqajeaibaqcLbmacaaIXaGaaGynaaaajugibiabgUcaRiaaiwdaca aI4aGaaG4naiaaiwdacaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaigdacaaIWaaaaKqzGeGaey4kaSIaaGymaiaaikdacaaIZa GaaGinaiaaiMdacaaI0aGaaGinaiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaI1aaaaKqzGeGaey4kaSIaaG4maiaaiIdacaaI3aGaaG imaiaaiEdacaaIYaGaaGimaaGccaGLOaGaayzkaaaabaqcfa4aaeWa aOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaigdaca aIWaaaaKqzGeGaey4kaSIaaGynaiaaikdacaaI4aGaeqiUdexcfa4a aWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaG ioaiaaiIdacaaIWaaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaaaaaaaa@9309@
γ= σ 2 μ 1 = θ 10 +528 θ 5 +2880 θ( θ 5 +24 )( θ 5 +120 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCjuaGdaahaaWcbeqc basaaKqzadGaaGOmaaaaaOqaaKqzGeGaeqiVd0wcfa4aaSbaaKqaGe aajugWaiaaigdaaSqabaqcfa4aaWbaaSqabeaajugibiadacUHYaIO aaaaaiabg2da9KqbaoaalaaakeaajugibiabeI7aXLqbaoaaCaaale qajeaibaqcLbmacaaIXaGaaGimaaaajugibiabgUcaRiaaiwdacaaI YaGaaGioaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaK qzGeGaey4kaSIaaGOmaiaaiIdacaaI4aGaaGimaaGcbaqcLbsacqaH 4oqCjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGynaaaajugibiabgUcaRiaaikdacaaI0aaakiaawIcacaGL PaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGynaaaajugibiabgUcaRiaaigdacaaIYaGaaGimaaGccaGL OaGaayzkaaaaaaaa@73C5@

The nature of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of Rani distribution have been shown graphically for varying values of parameter in Figure (3). The condition under which Rani distribution is over-dispersed, equi-dispersed, and under-dispersed along with condition under which Akash [3], Rama [9], Akshaya [10], Shanker [4], Amarendra [7], Aradhana [5], Sujatha [6], Devya [8], Lindley [1] and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed are presented in Table (2).


Figure 3: Graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of Rani distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Distribution

Over-dispersion ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWClmaaCaaajuaGbeqaaKqzadGaaGOm aaaaaKqbakaawIcacaGLPaaaaaa@3FBD@

Equi-dispersion ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabeaajugWaiaaikdaaaaa juaGcaGLOaGaayzkaaaaaa@3F26@

Under-dispersion ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabeaajugWaiaaikdaaaaa juaGcaGLOaGaayzkaaaaaa@3F26@

Rani

θ<2.449757591 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcqGH8aapcaaIYaGaaiOlaiaaisdacaaI0aGaaGyoaiaaiEdacaaI 1aGaaG4naiaaiwdacaaI5aGaaGymaaaa@416A@

θ=2.449757591 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcqGH9aqpcaaIYaGaaiOlaiaaisdacaaI0aGaaGyoaiaaiEdacaaI 1aGaaG4naiaaiwdacaaI5aGaaGymaaaa@416C@

θ>2.449757591 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcqGH+aGpcaaIYaGaaiOlaiaaisdacaaI0aGaaGyoaiaaiEdacaaI 1aGaaG4naiaaiwdacaaI5aGaaGymaaaa@416E@

Akash

θ<1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414D@

θ=1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414F@

θ>1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@4151@

Rama

θ<1.950164618 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI5aGaaGynaiaaicdacaaIXaGaaGOn aiaaisdacaaI2aGaaGymaiaaiIdaaaa@415D@

θ=1.950164618 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI5aGaaGynaiaaicdacaaIXaGaaGOn aiaaisdacaaI2aGaaGymaiaaiIdaaaa@415F@

θ>1.950164618 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI5aGaaGynaiaaicdacaaIXaGaaGOn aiaaisdacaaI2aGaaGymaiaaiIdaaaa@4161@

Akshaya

θ<1.327527885 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIZaGaaGOmaiaaiEdacaaI1aGaaGOm aiaaiEdacaaI4aGaaGioaiaaiwdaaaa@4164@

θ=1.950164618 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI5aGaaGynaiaaicdacaaIXaGaaGOn aiaaisdacaaI2aGaaGymaiaaiIdaaaa@415F@

θ>1.950164618 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI5aGaaGynaiaaicdacaaIXaGaaGOn aiaaisdacaaI2aGaaGymaiaaiIdaaaa@4161@

Shanker

θ<1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415A@

θ=1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415C@

Amarendra

θ<1.525763580 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI1aGaaGOmaiaaiwdacaaI3aGaaGOn aiaaiodacaaI1aGaaGioaiaaicdaaaa@415E@

θ=1.525763580 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aGaaGOmaiaaiwdacaaI3aGaaGOn aiaaiodacaaI1aGaaGioaiaaicdaaaa@4160@

θ>1.525763580 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI1aGaaGOmaiaaiwdacaaI3aGaaGOn aiaaiodacaaI1aGaaGioaiaaicdaaaa@4162@

Aradhana

θ<1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOm aiaaiAdacaaI1aGaaGimaiaaiwdaaaa@415C@

θ=1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOm aiaaiAdacaaI1aGaaGimaiaaiwdaaaa@415E@ >

θ>1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOm aiaaiAdacaaI1aGaaGimaiaaiwdaaaa@4160@

Sujatha

θ<1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4n aiaaigdacaaIXaGaaG4naiaaisdaaaa@4158@ >

θ=1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4n aiaaigdacaaIXaGaaG4naiaaisdaaaa@415A@

θ>1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4n aiaaigdacaaIXaGaaG4naiaaisdaaaa@415C@

Devya

θ<1.451669994 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI0aGaaGynaiaaigdacaaI2aGaaGOn aiaaiMdacaaI5aGaaGyoaiaaisdaaaa@416A@

θ=1.451669994 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI0aGaaGynaiaaigdacaaI2aGaaGOn aiaaiMdacaaI5aGaaGyoaiaaisdaaaa@416C@

θ>1.451669994 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI0aGaaGynaiaaigdacaaI2aGaaGOn aiaaiMdacaaI5aGaaGyoaiaaisdaaaa@416E@

Lindley

θ<1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@415E@

θ=1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4160@

θ>1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4162@

Exponential

θ<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaaaa@39F9@

θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaaaa@39FB@

θ>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaaaa@39FD@

Table 2: Over-dispersion, equi-dispersion and under-dispersion of Rani, Akash, Rama, Akshaya, Shanker, Amarendra, Aradhana, Sujatha, Devya, Lindley and exponential distributions for parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Hazard Rate Function and Mean Residual Life Function

Let f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B27@  and F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B07@  be the p.d.f. and c.d.f of a continuous random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ . The hazard rate function (also known as the failure rate function) and the mean residual life function of a continuous random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@36D3@ are, respectively, defined as

h( x )= lim Δx0 P( X<x+Δx|X>x ) Δx = f( x ) 1F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWfqaGcbaqcLbsaciGGSbGaaiyAaiaac2gaaSqaaKqzGe GaeyiLdqKaamiEaiabgkziUkaaicdaaSqabaqcfa4aaSaaaOqaaKqz GeGaamiuaKqbaoaabmaakeaajuaGdaabcaGcbaqcLbsacaWGybGaey ipaWJaamiEaiabgUcaRiabgs5aejaadIhacaaMc8oakiaawIa7aKqz GeGaamiwaiabg6da+iaadIhaaOGaayjkaiaawMcaaaqaaKqzGeGaey iLdqKaamiEaaaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGMbqcfa4a aeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaabaqcLbsacaaIXa GaeyOeI0IaamOraKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaa wMcaaaaaaaa@693C@  (3.1)

and  m( x )=E[ Xx|X>x ]= 1 1F( x ) x [ 1F( t ) ] dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGGa GaamyBaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaeyypa0JaamyraKqbaoaadmaakeaajuaGdaabcaGcbaqcLbsaca WGybGaeyOeI0IaamiEaaGccaGLiWoajugibiaadIfacqGH+aGpcaWG 4baakiaawUfacaGLDbaajugibiaaysW7cqGH9aqpcaaMe8Ecfa4aaS aaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIXaGaeyOeI0IaamOraKqb aoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaaajuaGdaWdXa Gcbaqcfa4aamWaaOqaaKqzGeGaaGymaiabgkHiTiaadAeajuaGdaqa daGcbaqcLbsacaWG0baakiaawIcacaGLPaaaaiaawUfacaGLDbaaaS qaaKqzGeGaamiEaaWcbaqcLbsacqGHEisPaiabgUIiYdGaaGjbVlaa ykW7caWGKbGaamiDaaaa@6BAB@  (3.2)

The corresponding hazard rate function, h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B29@ and the mean residual life function, m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B2E@ of the Rani distribution are obtained as

h( x )= θ 5 ( θ+ x 4 ) θ 4 x 4 +4 θ 3 x 3 +12 θ 2 x 2 +24θx+( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGynaaaajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaWG 4bqcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaaakiaawIcacaGLPa aaaeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaa aKqzGeGaamiEaKqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGe Gaey4kaSIaaGinaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI ZaaaaKqzGeGaamiEaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaK qzGeGaey4kaSIaaGymaiaaikdacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzad GaaGOmaaaajugibiabgUcaRiaaikdacaaI0aGaeqiUdeNaamiEaiab gUcaRKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaOGaayjkaiaa wMcaaaaaaaa@7C96@  (3.3)

and m( x )= 1 [ θ 4 x 4 +4 θ 3 x 3 +12 θ 2 x 2 +24θx+( θ 5 +24 ) ] e θx x [ θ 4 t 4 +4 θ 3 t 3 +12 θ 2 t 2 +24θt +( θ 5 +24 ) ] e θt dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajuaGdaWadaGcbaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibiaa dIhajuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibiabgUcaRi aaisdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugi biaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabgU caRiaaigdacaaIYaGaeqiUdexcfa4aaWbaaSqabKqaafaajug4aiaa ikdaaaqcLbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaa qcLbsacqGHRaWkcaaIYaGaaGinaiabeI7aXjaadIhacqGHRaWkjuaG daqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG ynaaaajugibiabgUcaRiaaikdacaaI0aaakiaawIcacaGLPaaaaiaa wUfacaGLDbaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaey OeI0IaeqiUdeNaamiEaaaaaaqcfa4aa8qCaOqaaKqbaoaadmaajugi bqaabeGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG inaaaajugibiaadshajuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaa jugibiabgUcaRiaaisdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzad GaaG4maaaajugibiaadshajuaGdaahaaWcbeqcbasaaKqzadGaaG4m aaaajugibiabgUcaRiaaigdacaaIYaGaeqiUdexcfa4aaWbaaSqabK qaGeaajugWaiaaikdaaaqcLbsacaWG0bqcfa4aaWbaaSqabKqaGeaa jugWaiaaikdaaaqcLbsacqGHRaWkcaaIYaGaaGinaiabeI7aXjaads haaOqaaKqzGeGaey4kaSscfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4a aWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaG inaaGccaGLOaGaayzkaaaaaiaawUfacaGLDbaajugibiaadwgajuaG daahaaWcbeqcbasaaKqzadGaeyOeI0IaeqiUdeNaamiDaaaajugibi aadsgacaWG0baaleaajugibiaadIhaaSqaaKqzGeGaeyOhIukacqGH RiI8aaaa@BE74@
= θ 4 x 4 +8 θ 3 x 3 +36 θ 2 x 2 +96θx+( θ 5 +120 ) θ[ θ 4 x 4 +4 θ 3 x 3 +12 θ 2 x 2 +24θx+( θ 5 +24 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaGinaaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaG inaaaajugibiabgUcaRiaaiIdacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaG4maaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzad GaaG4maaaajugibiabgUcaRiaaiodacaaI2aGaeqiUdexcfa4aaWba aSqabKqaGeaajugWaiaaikdaaaqcLbsacaWG4bqcfa4aaWbaaSqabK qaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaI5aGaaGOnaiabeI7a XjaadIhacqGHRaWkjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaa WcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaigdacaaIYaGa aGimaaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaGdaWadaGcba qcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugi biaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibiabgU caRiaaisdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaa jugibiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibi abgUcaRiaaigdacaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaikdaaaqcLbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcLbsacqGHRaWkcaaIYaGaaGinaiabeI7aXjaadIhacqGHRaWk juaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzad GaaGynaaaajugibiabgUcaRiaaikdacaaI0aaakiaawIcacaGLPaaa aiaawUfacaGLDbaaaaaaaa@9FBE@  (3.4)

It can be easily verified that h( 0 )= θ 5 θ 5 +24 =f( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaaGimaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGynaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaaacqGH9aqpca WGMbqcfa4aaeWaaOqaaKqzGeGaaGimaaGccaGLOaGaayzkaaaaaa@4FAB@ and m( 0 )= θ 5 +120 θ( θ 5 +24 ) = μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaaGimaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGynaaaajugibiabgUcaRiaaigdacaaIYaGaaGimaaGcbaqc LbsacqaH4oqCjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbe qcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaikdacaaI0aaakiaa wIcacaGLPaaaaaqcLbsacqGH9aqpcqaH8oqBjuaGdaWgaaqcbasaaK qzadGaaGymaaWcbeaajuaGdaahaaWcbeqaaKqzGeGamai4gkdiIcaa aaa@5CA7@ . It is also obvious from the graphs of h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B29@  and m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B2E@ that the shapes of h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B29@  is increasing, decreasing and upside bathtub, whereas the shapes of m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B2E@ is decreasing, increasing ( θ=0.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaK aaGfaajugibiabeI7aXjabg2da9iaaicdacaGGUaGaaGynaaqcaaMa ayjkaiaawMcaaaaa@3E55@  and downside bathtub. The graphs of the hazard rate function and mean residual life function of Rani distribution are shown in Figure (4).

Figure 4: Graphs of h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@3969@ and m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@396E@ of Rani distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Stochastic Orderings

Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@36D3@ is said to be smaller than a random variable Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@36D4@ in the

  1. stochastic order ( X st Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiwaiabgsMiJMqbaoaaBaaajeaibaqcLbmacaWGZbGa amiDaaWcbeaajugibiaadMfaaOGaayjkaiaawMcaaaaa@40B2@ if F X ( x ) F Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadIfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGHLjYScaWGgbqcfa4aaS baaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaaaaa@47BD@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aaaa@3782@
  2. hazard rate order ( X hr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiwaiabgsMiJMqbaoaaBaaajeaibaqcLbmacaWGObGa amOCaaWcbeaajugibiaadMfaaOGaayjkaiaawMcaaaaa@40A5@ if h X ( x ) h Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaSbaaKqaGeaajugWaiaadIfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGHLjYScaWGObqcfa4aaS baaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaaaaa@4801@  for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aaaa@3782@
  3. mean residual life order ( X mrl Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiwaiabgsMiJMqbaoaaBaaajeaibaqcLbmacaWGTbGa amOCaiaadYgaaSqabaqcLbsacaWGzbaakiaawIcacaGLPaaaaaa@419B@ if m X ( x ) m Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaSbaaKqaGeaajugWaiaadIfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGHKjYOcaWGTbqcfa4aaS baaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaaaaa@47FA@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aaaa@3782@
  4. likelihood ratio order ( X lr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiwaiabgsMiJMqbaoaaBaaajeaibaqcLbmacaWGSbGa amOCaaWcbeaajugibiaadMfaaOGaayjkaiaawMcaaaaa@40A9@ if f X ( x ) f Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamOzaKqbaoaaBaaajeaibaqcLbmacaWGybaaleqaaKqb aoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaqaaKqzGeGaam OzaKqbaoaaBaaajeaibaqcLbmacaWGzbaaleqaaKqbaoaabmaakeaa jugibiaadIhaaOGaayjkaiaawMcaaaaaaaa@46DF@  decreases in x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aaaa@3782@ .

The following results due to Shaked and Shanthikumar [11] are well known for establishing stochastic ordering of distributions

X lr YX hr YX mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKazba4=baqcLbmacaWGSbGaamOCaaqcbawa baqcLbsacaWGzbGaeyO0H4TaamiwaiabgsMiJMqbaoaaBaaajqwaa+ FaaKqzadGaamiAaiaadkhaaKqaGfqaaKqzGeGaamywaiabgkDiElaa dIfacqGHKjYOjuaGdaWgaaqcKfaG=haajugWaiaad2gacaWGYbGaam iBaaqcbawabaqcLbsacaWGzbaaaa@5A7E@ (4.1)
X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeaO qaaKqzGeGaey40H8naleaajugibiaadIfacqGHKjYOjuaGdaWgaaqc casaaKqzadGaam4CaiaadshaaWqabaqcLbsacaWGzbaaleqaaaaa@4253@

Rani distribution is ordered with respect to the strongest ‘likelihood ratio’ ordering as shown in the following theorem.

Theorem: Suppose X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8hpI4 haaa@3761@  Rani distributon ( θ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaa kiaawIcacaGLPaaaaaa@3D33@  and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb aaaa@3763@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8hpI4 haaa@3761@  Rani distribution ( θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaa kiaawIcacaGLPaaaaaa@3D34@ . If θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabg6da+iab eI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaaa@4123@ , then X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaadYgacaWGYbaaleqaaKqz GeGaamywaaaa@3E7E@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaadIgacaWGYbaaleqaaKqz GeGaamywaaaa@3E7A@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaad2gacaWGYbGaamiBaaWc beaajugibiaadMfaaaa@3F70@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaadohacaWG0baaleqaaKqz GeGaamywaaaa@3E87@ .
Proof: We have

f X ( x ) f Y ( x ) = θ 1 5 ( θ 2 5 +24 ) θ 2 5 ( θ 1 5 +24 ) ( θ 1 + x 4 θ 2 + x 4 ) e ( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSqaaS qaaKqzGeGaamOzaKqbaoaaBaaajiaibaqcLbmacaWGybaameqaaKqb aoaabmaaleaajugibiaadIhaaSGaayjkaiaawMcaaaqaaKqzGeGaam OzaKqbaoaaBaaajiaibaqcLbmacaWGzbaameqaaKqbaoaabmaaleaa jugibiaadIhaaSGaayjkaiaawMcaaaaajugibiabg2da9Kqbaoaala aakeaajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqa aKqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqbaoaabmaakeaaju gibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqbaoaa CaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaais daaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaa jugWaiaaikdaaSqabaqcfa4aaWbaaSqabKqaGeaajugWaiaaiwdaaa qcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaa igdaaSqabaqcfa4aaWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacq GHRaWkcaaIYaGaaGinaaGccaGLOaGaayzkaaaaaKqbaoaabmaakeaa juaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaG ymaaWcbeaajugibiabgUcaRiaadIhajuaGdaahaaWcbeqcbasaaKqz adGaaGinaaaaaOqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWai aaikdaaSqabaqcLbsacqGHRaWkcaWG4bqcfa4aaWbaaSqabKqaGeaa jugWaiaaisdaaaaaaaGccaGLOaGaayzkaaqcLbsacaWGLbqcfa4aaW baaSqabeaajugWaiabgkHiTKqbaoaabmaaleaajugabiabeI7aXLqb aoaaBaaajiaObaqcLbmacaaIXaaajiaObeaajugabiabgkHiTiabeI 7aXLqbaoaaBaaajiaObaqcLbmacaaIYaaajiaObeaaaSGaayjkaiaa wMcaaKqzGeGaamiEaaaaaaa@9F80@   ; x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4oaiaabc cacaWG4bGaeyOpa4JaaGimaaaa@3A17@

Now

ln f X ( x ) f Y ( x ) =ln[ θ 1 5 ( θ 2 5 +24 ) θ 2 5 ( θ 1 5 +24 ) ]+ln( θ 1 + x 4 θ 2 + x 4 )( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaKqbaoaaleaaleaajugibiaadAgajuaGdaWgaaqccasaaKqz adGaamiwaaadbeaajuaGdaqadaWcbaqcLbsacaWG4baaliaawIcaca GLPaaaaeaajugibiaadAgajuaGdaWgaaqccasaaKqzadGaamywaaad beaajuaGdaqadaWcbaqcLbsacaWG4baaliaawIcacaGLPaaaaaqcLb sacqGH9aqpciGGSbGaaiOBaKqbaoaadmaakeaajuaGdaWcaaGcbaqc LbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGda ahaaWcbeqcbasaaKqzadGaaGynaaaajuaGdaqadaGcbaqcLbsacqaH 4oqCjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaahaaWcbe qcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaikdacaaI0aaakiaa wIcacaGLPaaaaeaajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmaca aIYaaaleqaaKqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqbaoaa bmaakeaajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaale qaaKqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIa aGOmaiaaisdaaOGaayjkaiaawMcaaaaaaiaawUfacaGLDbaajugibi abgUcaRiGacYgacaGGUbqcfa4aaeWaaOqaaKqbaoaalaaakeaajugi biabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey 4kaSIaamiEaKqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaaGcbaqc LbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibi abgUcaRiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaaaaaa kiaawIcacaGLPaaajugibiabgkHiTKqbaoaabmaakeaajugibiabeI 7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0Ia eqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaakiaawIcaca GLPaaajugibiaadIhaaaa@A4DF@  .
This gives d dx { ln f X ( x ) f Y ( x ) }= 4( θ 1 θ 2 ) x 3 θ 2 + x 4 ( θ 1 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaaGcbaqcLbsacaWGKbGaamiEaaaajuaGdaGadaGc baqcLbsaciGGSbGaaiOBaKqbaoaaleaaleaajugibiaadAgajuaGda WgaaqccasaaKqzadGaamiwaaadbeaajuaGdaqadaWcbaqcLbsacaWG 4baaliaawIcacaGLPaaaaeaajugibiaadAgajuaGdaWgaaqccasaaK qzadGaamywaaadbeaajuaGdaqadaWcbaqcLbsacaWG4baaliaawIca caGLPaaaaaaakiaawUhacaGL9baajugibiabg2da9Kqbaoaalaaake aajugibiabgkHiTiaaisdajuaGdaqadaGcbaqcLbsacqaH4oqCjuaG daWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabgkHiTiabeI7aXL qbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaqc LbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaaakeaaju gibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGa ey4kaSIaamiEaKqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaaaaju gibiabgkHiTKqbaoaabmaakeaajugibiabeI7aXLqbaoaaBaaajeai baqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0IaeqiUdexcfa4aaSbaaK qaGeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaaaaa@7FF3@

Thus for θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabg6da+iab eI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaaa@4123@ , d dx { ln f X ( x ) f Y ( x ) }<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaaGcbaqcLbsacaWGKbGaamiEaaaajuaGdaGadaGc baqcLbsaciGGSbGaaiOBaKqbaoaaleaaleaajugibiaadAgajuaGda WgaaqccasaaKqzadGaamiwaaadbeaajuaGdaqadaWcbaqcLbsacaWG 4baaliaawIcacaGLPaaaaeaajugibiaadAgajuaGdaWgaaqccasaaK qzadGaamywaaadbeaajuaGdaqadaWcbaqcLbsacaWG4baaliaawIca caGLPaaaaaaakiaawUhacaGL9baajugibiabgYda8iaaicdaaaa@531B@ . This means that X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaadYgacaWGYbaaleqaaKqz GeGaamywaaaa@3E7E@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaadIgacaWGYbaaleqaaKqz GeGaamywaaaa@3E7A@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaad2gacaWGYbGaamiBaaWc beaajugibiaadMfaaaa@3F70@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyizImAcfa4aaSbaaKqaGeaajugWaiaadohacaWG0baaleqaaKqz GeGaamywaaaa@3E87@ .

Mean Deviations

The amount of scatter in a population is measured to some extent by the totality of deviations usually from mean and median. These are known as the mean deviation about the mean and the mean deviation about the median defined as
δ 1 ( X )= 0 | xμ | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehake aajuaGdaabdaGcbaqcLbsacaWG4bGaeyOeI0IaeqiVd0gakiaawEa7 caGLiWoaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYd GaaGPaVlaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGL PaaajugibiaadsgacaWG4baaaa@5700@  and δ 2 ( X )= 0 | xM | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehake aajuaGdaabdaGcbaqcLbsacaWG4bGaeyOeI0IaamytaaGccaGLhWUa ayjcSdaaleaajugibiaaicdaaSqaaKqzGeGaeyOhIukacqGHRiI8ai aaykW7caWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzk aaqcLbsacaWGKbGaamiEaaaa@561D@ , respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBcqGH9aqpcaWGfbqcfa4aaeWaaOqaaKqzGeGaamiwaaGccaGLOaGa ayzkaaaaaa@3DA2@  and M=Median ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb Gaeyypa0JaaeytaiaabwgacaqGKbGaaeyAaiaabggacaqGUbGaaeii aKqbaoaabmaajyaGbaqcLbsacaWGybaajyaGcaGLOaGaayzkaaaaaa@4301@ . The measures δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaaaaa@3E8E@ and δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaaaaa@3E8F@ can be calculated using the simplified relationships

δ 1 ( X )= 0 μ ( μx ) f( x )dx+ μ ( xμ ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehake aajuaGdaqadaGcbaqcLbsacqaH8oqBcqGHsislcaWG4baakiaawIca caGLPaaaaSqaaKqzGeGaaGimaaWcbaqcLbsacqaH8oqBaiabgUIiYd GaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhacqGHRaWkjuaGdaWdXbGcbaqcfa4aaeWaaOqaaK qzGeGaamiEaiabgkHiTiabeY7aTbGccaGLOaGaayzkaaaaleaajugi biabeY7aTbWcbaqcLbsacqGHEisPaiabgUIiYdGaamOzaKqbaoaabm aakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaamizaiaadIha aaa@69B8@
=μF( μ ) 0 μ xf( x )dx μ[ 1F( μ ) ]+ μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqaH8oqBcaWGgbqcfa4aaeWaaOqaaKqzGeGaeqiVd0gakiaawIca caGLPaaajugibiabgkHiTKqbaoaapehakeaajugibiaadIhacaaMc8 UaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhaaSqaaKqzGeGaaGimaaWcbaqcLbsacqaH8oqBai abgUIiYdGaeyOeI0IaeqiVd0wcfa4aamWaaOqaaKqzGeGaaGymaiab gkHiTiaadAeajuaGdaqadaGcbaqcLbsacqaH8oqBaOGaayjkaiaawM caaaGaay5waiaaw2faaKqzGeGaey4kaSscfa4aa8qCaOqaaKqzGeGa amiEaiaaykW7aSqaaKqzGeGaeqiVd0galeaajugibiabg6HiLcGaey 4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzk aaqcLbsacaWGKbGaamiEaaaa@7035@
=2μF( μ )2μ+2 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIYaGaeqiVd0MaamOraKqbaoaabmaakeaajugibiabeY7aTbGc caGLOaGaayzkaaqcLbsacqGHsislcaaIYaGaeqiVd0Maey4kaSIaaG OmaKqbaoaapehakeaajugibiaadIhacaaMc8oaleaajugibiabeY7a TbWcbaqcLbsacqGHEisPaiabgUIiYdGaamOzaKqbaoaabmaakeaaju gibiaadIhaaOGaayjkaiaawMcaaKqzGeGaamizaiaadIhaaaa@5636@
=2μF( μ )2 0 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIYaGaeqiVd0MaamOraKqbaoaabmaakeaajugibiabeY7aTbGc caGLOaGaayzkaaqcLbsacqGHsislcaaIYaqcfa4aa8qCaOqaaKqzGe GaamiEaiaaykW7aSqaaKqzGeGaaGimaaWcbaqcLbsacqaH8oqBaiab gUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawM caaKqzGeGaamizaiaadIhaaaa@522B@  (5.1)

and

δ 2 ( X )= 0 M ( Mx ) f( x )dx+ M ( xM ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehake aajuaGdaqadaGcbaqcLbsacaWGnbGaeyOeI0IaamiEaaGccaGLOaGa ayzkaaaaleaajugibiaaicdaaSqaaKqzGeGaamytaaGaey4kIipaca WGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsa caWGKbGaamiEaiabgUcaRKqbaoaapehakeaajuaGdaqadaGcbaqcLb sacaWG4bGaeyOeI0IaamytaaGccaGLOaGaayzkaaaaleaajugibiaa d2eaaSqaaKqzGeGaeyOhIukacqGHRiI8aiaadAgajuaGdaqadaGcba qcLbsacaWG4baakiaawIcacaGLPaaajugibiaadsgacaWG4baaaa@6629@
=MF( M ) 0 M xf( x )dx M[ 1F( M ) ]+ M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaWGnbGaaGPaVlaadAeajuaGdaqadaGcbaqcLbsacaWGnbaakiaa wIcacaGLPaaajugibiabgkHiTKqbaoaapehakeaajugibiaadIhaca aMc8UaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaSqaaKqzGeGaaGimaaWcbaqcLbsacaWGnb aacqGHRiI8aiabgkHiTiaad2eajuaGdaWadaGcbaqcLbsacaaIXaGa eyOeI0IaamOraKqbaoaabmaakeaajugibiaad2eaaOGaayjkaiaawM caaaGaay5waiaaw2faaKqzGeGaey4kaSscfa4aa8qCaOqaaKqzGeGa amiEaiaaykW7aSqaaKqzGeGaamytaaWcbaqcLbsacqGHEisPaiabgU IiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaaa@6C68@
=μ+2 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqGHsislcqaH8oqBcqGHRaWkcaaIYaqcfa4aa8qCaOqaaKqzGeGa amiEaiaaykW7aSqaaKqzGeGaamytaaWcbaqcLbsacqGHEisPaiabgU IiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaaa@4C5A@
=μ2 0 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqaH8oqBcqGHsislcaaIYaqcfa4aa8qCaOqaaKqzGeGaamiEaiaa ykW7aSqaaKqzGeGaaGimaaWcbaqcLbsacaWGnbaacqGHRiI8aiaadA gajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaajugibiaa dsgacaWG4baaaa@4AC1@  (5.2)

Using p.d.f. (1.1) and expression for the mean of Rani distribution (1.1), we get

0 μ x f( x )dx=μ { θ 5 ( μ 5 +θμ+1 )+5 θ 4 μ 4 +20 θ 3 μ 3 +60 θ 2 μ 2 +120( θμ+1 ) } e θμ θ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaiaaykW7aSqaaKqzGeGaaGimaaWcbaqcLbsacqaH 8oqBaiabgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaay jkaiaawMcaaKqzGeGaamizaiaadIhacqGH9aqpcqaH8oqBcqGHsisl juaGdaWcaaGcbaqcfa4aaiWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaS qabKqaGeaajugWaiaaiwdaaaqcfa4aaeWaaOqaaKqzGeGaeqiVd0wc fa4aaWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGHRaWkcqaH4o qCcqaH8oqBcqGHRaWkcaaIXaaakiaawIcacaGLPaaajugibiabgUca RiaaiwdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaaju gibiabeY7aTLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGa ey4kaSIaaGOmaiaaicdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzad GaaG4maaaajugibiabeY7aTLqbaoaaCaaaleqajeaibaqcLbmacaaI ZaaaaKqzGeGaey4kaSIaaGOnaiaaicdacqaH4oqCjuaGdaahaaWcbe qcbasaaKqzadGaaGOmaaaajugibiabeY7aTLqbaoaaCaaaleqajeai baqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGymaiaaikdacaaIWaqcfa 4aaeWaaOqaaKqzGeGaeqiUdeNaaGPaVlabeY7aTjabgUcaRiaaigda aOGaayjkaiaawMcaaaGaay5Eaiaaw2haaKqzGeGaamyzaKqbaoaaCa aaleqajeaibaqcLbmacqGHsislcqaH4oqCcaaMc8UaeqiVd0gaaaGc baqcLbsacqaH4oqCjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaa WcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaikdacaaI0aaa kiaawIcacaGLPaaaaaaaaa@A84D@  (5.3)
0 M x f( x )dx=μ { θ 5 ( M 5 +θM+1 )+5 θ 4 M 4 +20 θ 3 M 3 +60 θ 2 M 2 +120( θM+1 ) } e θM θ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaiaaykW7aSqaaKqzGeGaaGimaaWcbaqcLbsacaWG nbaacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawI cacaGLPaaajugibiaadsgacaWG4bGaeyypa0JaeqiVd0MaeyOeI0sc fa4aaSaaaOqaaKqbaoaacmaakeaajugibiabeI7aXLqbaoaaCaaale qajeaibaqcLbmacaaI1aaaaKqbaoaabmaakeaajugibiaad2eajuaG daahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiabeI7aXj aad2eacqGHRaWkcaaIXaaakiaawIcacaGLPaaajugibiabgUcaRiaa iwdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibi aad2eajuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibiabgUca RiaaikdacaaIWaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaio daaaqcLbsacaWGnbqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqc LbsacqGHRaWkcaaI2aGaaGimaiabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaIYaaaaKqzGeGaamytaKqbaoaaCaaaleqajeaibaqcLbma caaIYaaaaKqzGeGaey4kaSIaaGymaiaaikdacaaIWaqcfa4aaeWaaO qaaKqzGeGaeqiUdeNaaGPaVlaad2eacqGHRaWkcaaIXaaakiaawIca caGLPaaaaiaawUhacaGL9baajugibiaadwgajuaGdaahaaWcbeqcba saaKqzadGaeyOeI0IaeqiUdeNaaGPaVlaad2eaaaaakeaajugibiab eI7aXLqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaOGaayjkaiaa wMcaaaaaaaa@A12D@  (5.4)

Using expressions from (5.1), (5.2), (5.3), and (5.4), the mean deviation about mean, δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaaaaa@3E8E@  and the mean deviation about median, δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaaaaa@3E8F@  of Rani distribution (1.1) are obtained as

δ 1 ( X )= 2{ θ 4 μ 4 +8 θ 3 μ 3 +36 θ 2 μ 2 +96θμ+( θ 5 +120 ) } e θμ θ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiaaikdajuaGdaGadaGcbaqcLbsacqaH4oqCjuaGdaahaaWc beqcbasaaKqzadGaaGinaaaajugibiabeY7aTLqbaoaaCaaaleqaje aibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGioaiaaykW7cqaH4oqC juaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabeY7aTLqbao aaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaG4maiaa iAdacaaMc8UaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaa qcLbsacqaH8oqBjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugi biabgUcaRiaaiMdacaaI2aGaaGPaVlabeI7aXjaaykW7cqaH8oqBcq GHRaWkjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGynaaaajugibiabgUcaRiaaigdacaaIYaGaaGimaaGcca GLOaGaayzkaaaacaGL7bGaayzFaaqcLbsacaWGLbqcfa4aaWbaaSqa bKqaGeaajugWaiabgkHiTiabeI7aXjaaykW7cqaH8oqBaaaakeaaju gibiabeI7aXLqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqa jeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaOGaay jkaiaawMcaaaaaaaa@959E@  (5.5)
δ 2 ( X )= 2{ θ 5 ( M 5 +θM )+5 θ 4 M 4 +20 θ 3 M 3 +60 θ 2 M 2 +120θM+( θ 5 +120 ) } e θM θ( θ 5 +24 ) μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiaaikdajuaGdaGadaqcLbsaeaqabOqaaKqzGeGaeqiUdexc fa4aaWbaaSqabKqaGeaajugWaiaaiwdaaaqcfa4aaeWaaOqaaKqzGe GaamytaKqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4k aSIaeqiUdeNaamytaaGccaGLOaGaayzkaaqcLbsacqGHRaWkcaaI1a GaaGPaVlabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqz GeGaamytaKqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey 4kaSIaaGOmaiaaicdacaaMc8UaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaiodaaaqcLbsacaWGnbqcfa4aaWbaaSqabKqaGeaajugWai aaiodaaaqcLbsacqGHRaWkcaaI2aGaaGimaiaaykW7cqaH4oqCjuaG daahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiaad2eajuaGdaahaa WcbeqcbasaaKqzadGaaGOmaaaaaOqaaKqzGeGaey4kaSIaaGymaiaa ikdacaaIWaGaaGPaVlabeI7aXjaaykW7caWGnbGaey4kaSscfa4aae WaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiwda aaqcLbsacqGHRaWkcaaIXaGaaGOmaiaaicdaaOGaayjkaiaawMcaaa aacaGL7bGaayzFaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugW aiabgkHiTiabeI7aXjaaykW7caWGnbaaaaGcbaqcLbsacqaH4oqCju aGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGa aGynaaaajugibiabgUcaRiaaikdacaaI0aaakiaawIcacaGLPaaaaa qcLbsacqGHsislcqaH8oqBaaa@A8F6@  (5.6)

Bonferroni and Lorenz Curves

The Bonferroni and Lorenz curves [12] and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

B( p )= 1 pμ 0 q xf( x ) dx= 1 pμ [ 0 xf( x )dx q xf( x ) dx ]= 1 pμ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadchacqaH8o qBaaqcfa4aa8qCaOqaaKqzGeGaamiEaiaaykW7caWGMbqcfa4aaeWa aOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacaaMc8oaleaaju gibiaaicdaaSqaaKqzGeGaamyCaaGaey4kIipacaWGKbGaamiEaiab g2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamiCaiabeY 7aTbaajuaGdaWadaGcbaqcfa4aa8qCaOqaaKqzGeGaamiEaiaaykW7 caWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLb sacaWGKbGaamiEaiabgkHiTaWcbaqcLbsacaaIWaaaleaajugibiab g6HiLcGaey4kIipajuaGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaadA gajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaaaSqaaKqz GeGaamyCaaWcbaqcLbsacqGHEisPaiabgUIiYdGaaGPaVlaadsgaca WG4baakiaawUfacaGLDbaajugibiabg2da9Kqbaoaalaaakeaajugi biaaigdaaOqaaKqzGeGaamiCaiabeY7aTbaajuaGdaWadaGcbaqcLb sacqaH8oqBcqGHsisljuaGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaa dAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaaaSqaaK qzGeGaamyCaaWcbaqcLbsacqGHEisPaiabgUIiYdGaaGPaVlaadsga caWG4baakiaawUfacaGLDbaaaaa@9D6C@  (6.1)
and L( p )= 1 μ 0 q xf( x ) dx= 1 μ [ 0 xf( x )dx q xf( x ) dx ]= 1 μ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiabeY7aTbaaju aGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaadAgajuaGdaqadaGcbaqc LbsacaWG4baakiaawIcacaGLPaaaaSqaaKqzGeGaaGimaaWcbaqcLb sacaWGXbaacqGHRiI8aiaaykW7caWGKbGaamiEaiabg2da9Kqbaoaa laaakeaajugibiaaigdaaOqaaKqzGeGaeqiVd0gaaKqbaoaadmaake aajuaGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaadAgajuaGdaqadaGc baqcLbsacaWG4baakiaawIcacaGLPaaajugibiaadsgacaWG4bGaey OeI0caleaajugibiaaicdaaSqaaKqzGeGaeyOhIukacqGHRiI8aKqb aoaapehakeaajugibiaadIhacaaMc8UaamOzaKqbaoaabmaakeaaju gibiaadIhaaOGaayjkaiaawMcaaKqzGeGaaGPaVdWcbaqcLbsacaWG Xbaaleaajugibiabg6HiLcGaey4kIipacaWGKbGaamiEaaGccaGLBb GaayzxaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaa jugibiabeY7aTbaajuaGdaWadaGcbaqcLbsacqaH8oqBcqGHsislju aGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaadAgajuaGdaqadaGcbaqc LbsacaWG4baakiaawIcacaGLPaaaaSqaaKqzGeGaamyCaaWcbaqcLb sacqGHEisPaiabgUIiYdGaaGPaVlaadsgacaWG4baakiaawUfacaGL Dbaaaaa@9A97@  (6.2)

respectively or equivalently

B( p )= 1 pμ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadchacqaH8o qBaaqcfa4aa8qCaOqaaKqzGeGaamOraKqbaoaaCaaaleqajeaibaqc LbmacqGHsislcaaIXaaaaKqbaoaabmaakeaajugibiaadIhaaOGaay jkaiaawMcaaaWcbaqcLbsacaaIWaaaleaajugibiaadchaaiabgUIi YdGaaGPaVlaadsgacaWG4baaaa@53CF@  (6.3)
and L( p )= 1 μ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiabeY7aTbaaju aGdaWdXbGcbaqcLbsacaWGgbqcfa4aaWbaaSqabKqaGeaajugWaiab gkHiTiaaigdaaaqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaay zkaaaaleaajugibiaaicdaaSqaaKqzGeGaamiCaaGaey4kIipacaaM c8UaamizaiaadIhaaaa@52E4@  (6.4)

respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBcqGH9aqpcaWGfbqcfa4aaeWaaOqaaKqzGeGaamiwaaGccaGLOaGa ayzkaaaaaa@3DA2@  and q= F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb Gaeyypa0JaamOraKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaI XaaaaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaaaa@40B6@ .
The Bonferroni and Gini indices are thus defined as

B=1 0 1 B( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb Gaeyypa0JaaGymaiabgkHiTKqbaoaapehakeaajugibiaadkeajuaG daqadaGcbaqcLbsacaWGWbaakiaawIcacaGLPaaaaSqaaKqzGeGaaG imaaWcbaqcLbsacaaIXaaacqGHRiI8aiaaykW7caWGKbGaamiCaaaa @47FA@  (6.5)
and G=12 0 1 L( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaaGymaiabgkHiTiaaikdajuaGdaWdXbGcbaqcLbsacaWG mbqcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsaca aMc8oaleaajugibiaaicdaaSqaaKqzGeGaaGymaaGaey4kIipacaWG KbGaamiCaaaa@4954@  (6.6)

respectively.
Using p.d.f. of Rani distribution (1.1), we have

q xf( x ) dx= { θ 5 ( q 5 +θq )+5 θ 4 q 4 +20 θ 3 q 3 +60 θ 2 q 2 +120θq+( θ 5 +120 ) } e θq θ( θ 5 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaiaaykW7caWGMbqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaaaleaajugibiaadghaaSqaaKqzGeGaeyOhIu kacqGHRiI8aiaaykW7caWGKbGaamiEaiabg2da9Kqbaoaalaaakeaa juaGdaGadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzad GaaGynaaaajuaGdaqadaGcbaqcLbsacaWGXbqcfa4aaWbaaSqabKqa GeaajugWaiaaiwdaaaqcLbsacqGHRaWkcqaH4oqCcaaMc8UaamyCaa GccaGLOaGaayzkaaqcLbsacqGHRaWkcaaI1aGaaGPaVlabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaamyCaKqbaoaaCa aaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOmaiaaicda caaMc8UaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLb sacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacqGH RaWkcaaI2aGaaGimaiaaykW7cqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGOmaaaajugibiaaykW7caWGXbqcfa4aaWbaaSqabKqaGeaa jugWaiaaikdaaaqcLbsacqGHRaWkcaaIXaGaaGOmaiaaicdacaaMc8 UaeqiUdeNaaGPaVlaadghacqGHRaWkjuaGdaqadaGcbaqcLbsacqaH 4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRi aaigdacaaIYaGaaGimaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaqc LbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabeI7aXj aadghaaaaakeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeI7a XLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaG OmaiaaisdaaOGaayjkaiaawMcaaaaaaaa@AEE6@  (6.7)

Now using equation (6.7) in (6.1) and (6.2), we have

B( p )= 1 p [ 1 { θ 5 ( q 5 +θq )+5 θ 4 q 4 +20 θ 3 q 3 +60 θ 2 q 2 +120θq+( θ 5 +120 ) } e θq θ 5 +120 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadchaaaqcfa 4aamWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajuaGdaGa daGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaa aajuaGdaqadaGcbaqcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugW aiaaiwdaaaqcLbsacqGHRaWkcqaH4oqCcaaMc8UaamyCaaGccaGLOa GaayzkaaqcLbsacqGHRaWkcaaI1aGaaGPaVlabeI7aXLqbaoaaCaaa leqajeaibaqcLbmacaaI0aaaaKqzGeGaamyCaKqbaoaaCaaaleqaje aibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOmaiaaicdacaaMc8Ua eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacaWGXb qcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacqGHRaWkcaaI 2aGaaGimaiaaykW7cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG OmaaaajugibiaaykW7caWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaa ikdaaaqcLbsacqGHRaWkcaaIXaGaaGOmaiaaicdacaaMc8UaeqiUde NaaGPaVlaadghacqGHRaWkjuaGdaqadaGcbaqcLbsacqaH4oqCjuaG daahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaigdaca aIYaGaaGimaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaqcLbsacaWG Lbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabeI7aXjaadghaaa aakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaa aKqzGeGaey4kaSIaaGymaiaaikdacaaIWaaaaaGccaGLBbGaayzxaa aaaa@A75E@  (6.8)
and L( p )=1 { θ 5 ( q 5 +θq )+5 θ 4 q 4 +20 θ 3 q 3 +60 θ 2 q 2 +120θq+( θ 5 +120 ) } e θq θ 5 +120 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqbaoaacmaakeaajugibi abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqbaoaabmaa keaajugibiaadghajuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaaju gibiabgUcaRiabeI7aXjaaykW7caWGXbaakiaawIcacaGLPaaajugi biabgUcaRiaaiwdacaaMc8UaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaisdaaaqcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaa isdaaaqcLbsacqGHRaWkcaaIYaGaaGimaiaaykW7cqaH4oqCjuaGda ahaaWcbeqcbasaaKqzadGaaG4maaaajugibiaadghajuaGdaahaaWc beqcbasaaKqzadGaaG4maaaajugibiabgUcaRiaaiAdacaaIWaGaaG PaVlabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGa aGPaVlaadghajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibi abgUcaRiaaigdacaaIYaGaaGimaiaaykW7cqaH4oqCcaaMc8UaamyC aiabgUcaRKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGymaiaaikdacaaIWaaa kiaawIcacaGLPaaaaiaawUhacaGL9baajugibiaadwgajuaGdaahaa WcbeqcbasaaKqzadGaeyOeI0IaeqiUdeNaamyCaaaaaOqaaKqzGeGa eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGHRa WkcaaIXaGaaGOmaiaaicdaaaaaaa@A0C5@ (6.9)

Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini indices are obtained as

B=1 { θ 5 ( q 5 +θq )+5 θ 4 q 4 +20 θ 3 q 3 +60 θ 2 q 2 +120θq+( θ 5 +120 ) } e θq θ 5 +120 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb Gaeyypa0JaaGymaiabgkHiTKqbaoaalaaakeaajuaGdaGadaGcbaqc LbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajuaGda qadaGcbaqcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaiwda aaqcLbsacqGHRaWkcqaH4oqCcaaMc8UaamyCaaGccaGLOaGaayzkaa qcLbsacqGHRaWkcaaI1aGaaGPaVlabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaI0aaaaKqzGeGaamyCaKqbaoaaCaaaleqajeaibaqcLb macaaI0aaaaKqzGeGaey4kaSIaaGOmaiaaicdacaaMc8UaeqiUdexc fa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacaWGXbqcfa4aaW baaSqabKqaGeaajugWaiaaiodaaaqcLbsacqGHRaWkcaaI2aGaaGim aiaaykW7cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaju gibiaaykW7caWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqc LbsacqGHRaWkcaaIXaGaaGOmaiaaicdacaaMc8UaeqiUdeNaaGPaVl aadghacqGHRaWkjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWc beqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaigdacaaIYaGaaG imaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaqcLbsacaWGLbqcfa4a aWbaaSqabKqaGeaajugWaiabgkHiTiabeI7aXjaadghaaaaakeaaju gibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGa ey4kaSIaaGymaiaaikdacaaIWaaaaaaa@9C7D@  (6.10)
G= 2{ θ 5 ( q 5 +θq )+5 θ 4 q 4 +20 θ 3 q 3 +60 θ 2 q 2 +120θq+( θ 5 +120 ) } e θq θ 5 +120 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmaKqbaoaacmaakeaajugi biabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqbaoaabm aakeaajugibiaadghajuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaa jugibiabgUcaRiabeI7aXjaaykW7caWGXbaakiaawIcacaGLPaaaju gibiabgUcaRiaaiwdacaaMc8UaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaisdaaaqcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugWai aaisdaaaqcLbsacqGHRaWkcaaIYaGaaGimaiaaykW7cqaH4oqCjuaG daahaaWcbeqcbasaaKqzadGaaG4maaaajugibiaadghajuaGdaahaa WcbeqcbasaaKqzadGaaG4maaaajugibiabgUcaRiaaiAdacaaIWaGa aGPaVlabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGe GaaGPaVlaadghajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugi biabgUcaRiaaigdacaaIYaGaaGimaiaaykW7cqaH4oqCcaaMc8Uaam yCaiabgUcaRKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqa jeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGymaiaaikdacaaIWa aakiaawIcacaGLPaaaaiaawUhacaGL9baajugibiaadwgajuaGdaah aaWcbeqcbasaaKqzadGaeyOeI0IaeqiUdeNaamyCaaaaaOqaaKqzGe GaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGH RaWkcaaIXaGaaGOmaiaaicdaaaGaeyOeI0IaaGymaaaa@9DCD@  (6.11)

Order Statistics and Renyi Entropy Measure

Distribution of Order Statistics

 Let X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb qcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacaGGSaGaaGPa VlaadIfajuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiaacY cacaaMc8UaaiOlaiaac6cacaGGUaGaaiilaiaaykW7caWGybqcfa4a aSbaaKqaGeaajugWaiaad6gaaSqabaaaaa@4BA1@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@  from Rani distribution (1.1). Let X ( 1 ) < X ( 2 ) <...< X ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb qcfa4aaSbaaKqaGeaalmaabmaajeaibaqcLbmacaaIXaaajeaicaGL OaGaayzkaaaaleqaaKqzGeGaeyipaWJaamiwaKqbaoaaBaaajeaiba WcdaqadaqcbasaaKqzadGaaGOmaaqcbaIaayjkaiaawMcaaaWcbeaa jugibiabgYda8iaaykW7caaMc8UaaiOlaiaac6cacaGGUaGaaGPaVl aaykW7cqGH8aapcaWGybqcfa4aaSbaaKqaGeaalmaabmaajeaibaqc LbmacaWGUbaajeaicaGLOaGaayzkaaaaleqaaaaa@53E0@ denote the corresponding order statistics. The p.d.f. and the c.d.f. of the k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ th order statistic, say Y= X ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb Gaeyypa0JaamiwaKqbaoaaBaaajeaibaWcdaqadaqcbasaaKqzadGa am4AaaqcbaIaayjkaiaawMcaaaWcbeaaaaa@3E30@ are given by

f Y ( y )= n! ( k1 )!( nk )! F k1 ( y ) { 1F( y ) } nk f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamyEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcba qcLbsacaWGUbGaaiyiaaGcbaqcfa4aaeWaaOqaaKqzGeGaam4Aaiab gkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaaiyiaiaaykW7juaGda qadaGcbaqcLbsacaWGUbGaeyOeI0Iaam4AaaGccaGLOaGaayzkaaqc LbsacaGGHaaaaiaaykW7caWGgbqcfa4aaWbaaSqabKqaGeaajugWai aadUgacqGHsislcaaIXaaaaKqbaoaabmaakeaajugibiaadMhaaOGa ayjkaiaawMcaaKqbaoaacmaakeaajugibiaaigdacqGHsislcaWGgb qcfa4aaeWaaOqaaKqzGeGaamyEaaGccaGLOaGaayzkaaaacaGL7bGa ayzFaaqcfa4aaWbaaSqabKqaGeaajugWaiaad6gacqGHsislcaWGRb aaaKqzGeGaamOzaKqbaoaabmaakeaajugibiaadMhaaOGaayjkaiaa wMcaaaaa@6FA6@

= n! ( k1 )!( nk )! l=0 nk ( nk l ) ( 1 ) l F k+l1 ( y )f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaWGUbGaaiyiaaGcbaqcfa4aaeWaaOqa aKqzGeGaam4AaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaai yiaiaaykW7juaGdaqadaGcbaqcLbsacaWGUbGaeyOeI0Iaam4AaaGc caGLOaGaayzkaaqcLbsacaGGHaaaaiaaykW7juaGdaaeWbGcbaqcfa 4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsacaWGUbGaeyOeI0Ia am4AaaGcbaqcLbsacaWGSbaaaaGccaGLOaGaayzkaaaaleaajugibi aadYgacqGH9aqpcaaIWaaaleaajugibiaad6gacqGHsislcaWGRbaa cqGHris5aKqbaoaabmaakeaajugibiabgkHiTiaaigdaaOGaayjkai aawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaWGSbaaaKqzGeGaamOr aKqbaoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaSIaamiBaiabgk HiTiaaigdaaaqcfa4aaeWaaOqaaKqzGeGaamyEaaGccaGLOaGaayzk aaqcLbsacaWGMbqcfa4aaeWaaOqaaKqzGeGaamyEaaGccaGLOaGaay zkaaaaaa@7417@

and

F Y ( y )= j=k n ( n j ) F j ( y ) { 1F( y ) } nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKazba4=baqcLbmacaWGzbaajeaybeaajuaGdaqadaqc aawaaKqzGeGaamyEaaqcaaMaayjkaiaawMcaaKqzGeGaeyypa0tcfa 4aaabCaKaaGfaajuaGdaqadaqcaawaaKqzGeqbaeqabiqaaaqcaawa aKqzGeGaamOBaaqcaawaaKqzGeGaamOAaaaaaKaaGjaawIcacaGLPa aaaKqaGfaajugibiaadQgacqGH9aqpcaWGRbaajeaybaqcLbsacaWG UbaacqGHris5aiaaykW7caWGgbqcfa4aaWbaaKqaGfqajqwaa+FaaK qzadGaamOAaaaajuaGdaqadaqcaawaaKqzGeGaamyEaaqcaaMaayjk aiaawMcaaKqbaoaacmaajaaybaqcLbsacaaIXaGaeyOeI0IaamOraK qbaoaabmaajaaybaqcLbsacaWG5baajaaycaGLOaGaayzkaaaacaGL 7bGaayzFaaqcfa4aaWbaaKqaGfqajqwaa+FaaKqzadGaamOBaiabgk HiTiaadQgaaaaaaa@7043@

= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l F j+l ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaaeWbGcbaqcfa4aaabCaOqaaKqbaoaabmaakeaajugibuaa beqaceaaaOqaaKqzGeGaamOBaaGcbaqcLbsacaWGQbaaaaGccaGLOa GaayzkaaaaleaajugibiaadYgacqGH9aqpcaaIWaaaleaajugibiaa d6gacqGHsislcaWGQbaacqGHris5aKqbaoaabmaakeaajugibuaabe qaceaaaOqaaKqzGeGaamOBaiabgkHiTiaadQgaaOqaaKqzGeGaamiB aaaaaOGaayjkaiaawMcaaaWcbaqcLbsacaWGQbGaeyypa0Jaam4Aaa WcbaqcLbsacaWGUbaacqGHris5aiaaykW7juaGdaqadaGcbaqcLbsa cqGHsislcaaIXaaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaK qzadGaamiBaaaajugibiaadAeajuaGdaahaaWcbeqcbasaaKqzadGa amOAaiabgUcaRiaadYgaaaqcfa4aaeWaaOqaaKqzGeGaamyEaaGcca GLOaGaayzkaaaaaa@693B@ ,

respectively, for k=1,2,3,...,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb Gaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOl aiaac6cacaGGUaGaaiilaiaad6gaaaa@4078@ .

 Thus, the p.d.f. and the c.d.f of k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ th order statistics of Rani distribution are given by

f Y ( y )= n! θ 5 ( θ+ x 4 ) e θx ( θ 5 +24 )( k1 )!( nk )! l=0 nk ( nk l ) ( 1 ) l × [ 1{ 1+ θx( θ 3 x 3 +4 θ 2 x 2 +12θx+24 ) θ 5 +24 } e θx ] k+l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamyEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcba qcLbsacaWGUbGaaiyiaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbma caaI1aaaaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaykW7ca WG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaaakiaawIcacaGL PaaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0Iaeq iUdeNaamiEaaaaaOqaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaa CaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaais daaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiaadUgacqGHsisl caaIXaaakiaawIcacaGLPaaajugibiaacgcacaaMc8Ecfa4aaeWaaO qaaKqzGeGaamOBaiabgkHiTiaadUgaaOGaayjkaiaawMcaaKqzGeGa aiyiaaaacaaMc8Ecfa4aaabCaOqaaKqbaoaabmaakeaajugibuaabe qaceaaaOqaaKqzGeGaamOBaiabgkHiTiaadUgaaOqaaKqzGeGaamiB aaaaaOGaayjkaiaawMcaaaWcbaqcLbsacaWGSbGaeyypa0JaaGimaa WcbaqcLbsacaWGUbGaeyOeI0Iaam4AaaGaeyyeIuoajuaGdaqadaGc baqcLbsacqGHsislcaaIXaaakiaawIcacaGLPaaajuaGdaahaaWcbe qcbasaaKqzadGaamiBaaaajugibiabgEna0Mqbaoaadmaakeaajugi biaaigdacqGHsisljuaGdaGadaGcbaqcLbsacaaIXaGaey4kaSscfa 4aaSaaaOqaaKqzGeGaeqiUdeNaamiEaKqbaoaabmaakeaajugibiab eI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaamiEaK qbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGin aiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaam iEaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIa aGymaiaaikdacqaH4oqCcaWG4bGaey4kaSIaaGOmaiaaisdaaOGaay jkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaaaaOGaay5Eaiaaw2 haaKqzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH 4oqCcaWG4baaaaGccaGLBbGaayzxaaqcfa4aaWbaaSqabKqaGeaaju gWaiaadUgacqGHRaWkcaWGSbGaeyOeI0IaaGymaaaaaaa@D0AB@

and

F Y ( y )= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l [ 1{ 1+ θx( θ 3 x 3 +4 θ 2 x 2 +12θx+24 ) θ 5 +24 } e θx ] j+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamyEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaaeWbGcba qcfa4aaabCaOqaaKqbaoaabmaakeaajugibuaabeqaceaaaOqaaKqz GeGaamOBaaGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaaaleaaju gibiaadYgacqGH9aqpcaaIWaaaleaajugibiaad6gacqGHsislcaWG QbaacqGHris5aKqbaoaabmaakeaajugibuaabeqaceaaaOqaaKqzGe GaamOBaiabgkHiTiaadQgaaOqaaKqzGeGaamiBaaaaaOGaayjkaiaa wMcaaaWcbaqcLbsacaWGQbGaeyypa0Jaam4AaaWcbaqcLbsacaWGUb aacqGHris5aiaaykW7juaGdaqadaGcbaqcLbsacqGHsislcaaIXaaa kiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaamiBaaaaju aGdaWadaGcbaqcLbsacaaIXaGaeyOeI0scfa4aaiWaaOqaaKqzGeGa aGymaiabgUcaRKqbaoaalaaakeaajugibiabeI7aXjaadIhajuaGda qadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4m aaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaaju gibiabgUcaRiaaisdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGa aGOmaaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaa aajugibiabgUcaRiaaigdacaaIYaGaeqiUdeNaamiEaiabgUcaRiaa ikdacaaI0aaakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbaoaaCa aaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisda aaaakiaawUhacaGL9baajugibiaadwgajuaGdaahaaWcbeqcbasaaK qzadGaeyOeI0IaeqiUdeNaamiEaaaaaOGaay5waiaaw2faaKqbaoaa CaaaleqajeaibaqcLbmacaWGQbGaey4kaSIaamiBaaaaaaa@A7AF@

Renyi Entropy Measure

Entropy of a random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy [13]. If X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ is a continuous random variable having probability density function f( . ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaaiOlaaGccaGLOaGaayzkaaaaaa@3ADC@ , then Renyi entropy is defined as

T R ( γ )= 1 1γ log{ f γ ( x )dx } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaKqaGeaajugWaiaadkfaaSqabaqcfa4aaeWaaOqaaKqz GeGaeq4SdCgakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGymaiabgkHiTiabeo7aNbaaciGG SbGaai4BaiaacEgajuaGdaGadaGcbaqcfa4aa8qaaOqaaKqzGeGaam OzaKqbaoaaCaaaleqajeaibaqcLbmacqaHZoWzaaqcfa4aaeWaaOqa aKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacaWGKbGaamiEaaWcbe qabKqzGeGaey4kIipaaOGaay5Eaiaaw2haaaaa@5A2F@

where γ>0andγ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH+aGpcaaIWaGaaGPaVlaaykW7caaMc8Uaaeyyaiaab6gacaqG KbGaaGPaVlaaykW7caaMc8Uaeq4SdCMaeyiyIKRaaGymaaaa@4A15@ .
Thus, the Renyi entropy for the Rani distribution (1.1) is obtained as

T R ( γ )= 1 1γ log[ 0 θ 5 γ ( θ 5 +24 ) γ ( θ+ x 4 ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaKqaGeaajugWaiaadkfaaSqabaqcfa4aaeWaaOqaaKqz GeGaeq4SdCgakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGymaiabgkHiTiabeo7aNbaaciGG SbGaai4BaiaacEgajuaGdaWadaGcbaqcfa4aa8qCaOqaaKqbaoaala aakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaa aKqbaoaaCaaaleqajeaibaqcLbmacqaHZoWzaaaakeaajuaGdaqada GcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaa jugibiabgUcaRiaaikdacaaI0aaakiaawIcacaGLPaaajuaGdaahaa WcbeqcbasaaKqzadGaeq4SdCgaaaaajuaGdaqadaGcbaqcLbsacqaH 4oqCcqGHRaWkcaaMc8UaamiEaKqbaoaaCaaaleqajeaibaqcLbmaca aI0aaaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiab eo7aNbaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0 IaeqiUdeNaaGPaVlabeo7aNjaaykW7caWG4baaaKqzGeGaamizaiaa dIhaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYdaaki aawUfacaGLDbaaaaa@85F0@

= 1 1γ log[ 0 θ 6 γ ( θ 5 +24 ) γ ( 1+ x 4 θ ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaape hakeaajuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOnaaaajuaGdaahaaWcbeqcbasaaKqzadGaeq4SdCgaaa Gcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaGccaGLOaGaay zkaaqcfa4aaWbaaSqabKqaGeaajugWaiabeo7aNbaaaaqcfa4aaeWa aOqaaKqzGeGaaGymaiabgUcaRKqbaoaalaaakeaajugibiaadIhaju aGdaahaaWcbeqcbasaaKqzadGaaGinaaaaaOqaaKqzGeGaeqiUdeha aaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiabeo7aNb aajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaeqiU deNaaGPaVlabeo7aNjaaykW7caWG4baaaKqzGeGaamizaiaadIhaaS qaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYdaakiaawUfa caGLDbaaaaa@7E3F@

= 1 1γ log[ 0 θ 6 γ ( θ 5 +24 ) γ j=0 ( γ j ) ( x 4 θ ) j e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaape hakeaajuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOnaaaajuaGdaahaaWcbeqcbasaaKqzadGaeq4SdCgaaa Gcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaGccaGLOaGaay zkaaqcfa4aaWbaaSqabKqaGeaajugWaiabeo7aNbaaaaqcfa4aaabC aOqaaKqbaoaabmaakeaajugibuaabeqaceaaaOqaaKqzGeGaeq4SdC gakeaajugibiaadQgaaaaakiaawIcacaGLPaaaaSqaaKqzGeGaamOA aiabg2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5aiaaykW7ju aGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGaamiEaKqbaoaaCaaaleqa baqcLbsacaaI0aaaaaGcbaqcLbsacqaH4oqCaaaakiaawIcacaGLPa aajuaGdaahaaWcbeqcbasaaKqzadGaamOAaaaajugibiaadwgajuaG daahaaWcbeqcbasaaKqzadGaeyOeI0IaeqiUdeNaaGPaVlabeo7aNj aaykW7caWG4baaaKqzGeGaamizaiaadIhaaSqaaKqzGeGaaGimaaWc baqcLbsacqGHEisPaiabgUIiYdaakiaawUfacaGLDbaaaaa@8AC2@

= 1 1γ log[ j=0 ( γ j ) θ 6 γj ( θ 5 +24 ) γ 0 e θγx x 4j+11 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaaqa hakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiabeo7a NbGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaaaleaajugibiaadQ gacqGH9aqpcaaIWaaaleaajugibiabg6HiLcGaeyyeIuoajuaGdaWc aaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOnaa aajuaGdaahaaWcbeqcbasaaKqzadGaeq4SdCMaeyOeI0IaamOAaaaa aOqaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaOGaayjkaiaa wMcaaKqbaoaaCaaaleqajeaibaqcLbmacqaHZoWzaaaaaKqbaoaape hakeaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0Ia eqiUdeNaaGPaVlabeo7aNjaaykW7caWG4baaaaWcbaqcLbsacaaIWa aaleaajugibiabg6HiLcGaey4kIipacaWG4bqcfa4aaWbaaSqabKqa GeaajugWaiaaisdacaWGQbGaey4kaSIaaGymaiabgkHiTiaaigdaaa qcLbsacaWGKbGaamiEaaGccaGLBbGaayzxaaaaaa@875D@

= 1 1γ log[ j=0 ( γ j ) θ 6 γj ( θ 5 +24 ) γ Γ( 4j+1 ) ( θγ ) 4j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaaqa hakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiabeo7a NbGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaaaleaajugibiaadQ gacqGH9aqpcaaIWaaaleaajugibiabg6HiLcGaeyyeIuoajuaGdaWc aaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOnaa aajuaGdaahaaWcbeqcbasaaKqzadGaeq4SdCMaeyOeI0IaamOAaaaa aOqaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaOGaayjkaiaa wMcaaKqbaoaaCaaaleqajeaibaqcLbmacqaHZoWzaaaaaKqbaoaala aakeaajugibiabfo5ahLqbaoaabmaakeaajugibiaaisdacaWGQbGa ey4kaSIaaGymaaGccaGLOaGaayzkaaaabaqcfa4aaeWaaOqaaKqzGe GaeqiUdeNaeq4SdCgakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasa aKqzadGaaGinaiaadQgacqGHRaWkcaaIXaaaaaaaaOGaay5waiaaw2 faaaaa@7ECC@

= 1 1γ log[ j=0 ( γ j ) θ γ1 ( 1+θ ) γj ( θ+2 ) γ Γ( 4j+1 ) ( γ ) j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaaGymaaqaaiaaigdacqGHsislcqaHZoWzaaGaciiBaiaa c+gacaGGNbWaamWaaeaadaaeWbqaamaabmaabaqbaeqabiqaaaqaai abeo7aNbqaaiaadQgaaaaacaGLOaGaayzkaaaajyaGbaGaamOAaiab g2da9iaaicdaaeaacqGHEisPaKqbakabggHiLdWaaSaaaeaacqaH4o qCdaahaaqabKGbagaacqaHZoWzcqGHsislcaaIXaaaaKqbaoaabmaa baGaaGymaiabgUcaRiabeI7aXbGaayjkaiaawMcaamaaCaaabeqcga yaaiabeo7aNjabgkHiTiaadQgaaaaajuaGbaWaaeWaaeaacqaH4oqC cqGHRaWkcaaIYaaacaGLOaGaayzkaaqcga4aaWbaaeqabaGaeq4SdC gaaaaajuaGdaWcaaqaaiabfo5ahnaabmaabaGaaGinaiaadQgacqGH RaWkcaaIXaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaHZoWzaiaawI cacaGLPaaadaahaaqabKGbagaacaWGQbGaey4kaSIaaGymaaaaaaaa juaGcaGLBbGaayzxaaaaaa@7147@
= 1 1γ log[ j=0 ( γ j ) θ 6 γ5j1 ( θ 5 +24 ) γ Γ( 4j+1 ) ( γ ) 4j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaaqa hakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiabeo7a NbGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaaaleaajugibiaadQ gacqGH9aqpcaaIWaaaleaajugibiabg6HiLcGaeyyeIuoajuaGdaWc aaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOnaa aajuaGdaahaaWcbeqcbasaaKqzadGaeq4SdCMaeyOeI0IaaGynaiaa dQgacqGHsislcaaIXaaaaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUde xcfa4aaWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGHRaWkcaaI YaGaaGinaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWai abeo7aNbaaaaqcfa4aaSaaaOqaaKqzGeGaeu4KdCucfa4aaeWaaOqa aKqzGeGaaGinaiaadQgacqGHRaWkcaaIXaaakiaawIcacaGLPaaaae aajuaGdaqadaGcbaqcLbsacqaHZoWzaOGaayjkaiaawMcaaKqbaoaa CaaaleqajeaibaqcLbmacaaI0aGaamOAaiabgUcaRiaaigdaaaaaaa GccaGLBbGaayzxaaaaaa@7F7D@

Stress-Strength Reliability

The stress-strength reliability gives the idea about the life of a component which has random strength X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ that is subjected to a random stress Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb aaaa@3763@ . When the stress applied to it exceeds the strength, the component fails instantly and the component will function satisfactorily till X>Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyOpa4Jaamywaaaa@3948@ . Therefore, R=P( Y<X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb Gaeyypa0JaamiuaKqbaoaabmaakeaajugibiaadMfacqGH8aapcaWG ybaakiaawIcacaGLPaaaaaa@3EB0@ is a measure of the component reliability and in statistical literature it is known as stress-strength parameter. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.

Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb aaaa@3763@ be independent strength and stress random variables having Rani distribution (1.1) with parameter θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaaaaa@3B08@  and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@3B09@  respectively. Then the stress-strength reliability R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@36CD@ of Rani distribution can be obtained as

R=P( Y<X )= 0 P( Y<X|X=x ) f X ( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb Gaeyypa0JaamiuaKqbaoaabmaakeaajugibiaadMfacqGH8aapcaWG ybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehakeaajugibi aadcfajuaGdaqadaGcbaqcLbsacaWGzbGaeyipaWJaamiwaiaacYha caWGybGaeyypa0JaamiEaaGccaGLOaGaayzkaaaaleaajugibiaaic daaSqaaKqzGeGaeyOhIukacqGHRiI8aiaadAgajuaGdaWgaaqcbasa aKqzadGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4baakiaawI cacaGLPaaajugibiaadsgacaWG4baaaa@5B50@
= 0 f( x; θ 1 ) F( x; θ 2 )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWdXbGcbaqcLbsacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiE aiaacUdacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaaaO GaayjkaiaawMcaaaWcbaqcLbsacaaIWaaaleaajugibiabg6HiLcGa ey4kIipacaaMc8UaaGPaVlaadAeajuaGdaqadaGcbaqcLbsacaWG4b Gaai4oaiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaGc caGLOaGaayzkaaqcLbsacaWGKbGaamiEaaaa@5796@
=1 θ 1 5 [ 40320 θ 2 4 +20160 θ 2 3 ( θ 1 + θ 2 )+8640 θ 2 2 ( θ 1 + θ 2 ) 2 +2880 θ 2 ( θ 1 + θ 2 ) 3 +24( θ 2 5 + θ 1 θ 2 4 +24 ) ( θ 1 + θ 2 ) 4 +24 θ 1 θ 2 3 ( θ 1 + θ 2 ) 5 +24 θ 1 θ 2 2 ( θ 1 + θ 2 ) 6 +24 θ 1 θ 2 ( θ 1 + θ 2 ) 7 + θ 1 ( θ 2 5 +24 ) ( θ 1 + θ 2 ) 8 ] ( θ 1 5 +24 )( θ 2 5 +24 ) ( θ 1 + θ 2 ) 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaSba aKqaGeaajugWaiaaigdaaSqabaqcfa4aaWbaaSqabKqaGeaajugWai aaiwdaaaqcfa4aamWaaKqzGeabaeqakeaajugibiaaisdacaaIWaGa aG4maiaaikdacaaIWaGaaGPaVlabeI7aXLqbaoaaBaaajeaibaqcLb macaaIYaaaleqaaKqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqz GeGaey4kaSIaaGOmaiaaicdacaaIXaGaaGOnaiaaicdacaaMc8Uaeq iUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4aaWbaaSqa bKqaGeaajugWaiaaiodaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa 4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcqaH4oqC juaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaOGaayjkaiaawMcaaK qzGeGaey4kaSIaaGioaiaaiAdacaaI0aGaaGimaiaaykW7cqaH4oqC juaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaahaaWcbeqcba saaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaWg aaqcbasaaKqzadGaaGymaaWcbeaajugibiabgUcaRiabeI7aXLqbao aaBaaajeaibaqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaqcfa4a aWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIYaGaaG ioaiaaiIdacaaIWaGaaGPaVlabeI7aXLqbaoaaBaaajeaibaqcLbma caaIYaaaleqaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaBaaaje aibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaeqiUdexcfa4aaSba aKqaGeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaajuaGdaahaa WcbeqcbasaaKqzadGaaG4maaaaaOqaaKqzGeGaey4kaSIaaGOmaiaa isdajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqzad GaaGOmaaWcbeaajuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugi biabgUcaRiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaK qzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4a aWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaIYaGaaG inaaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4a aSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcqaH4oqCju aGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaOGaayjkaiaawMcaaKqb aoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOmai aaisdacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugi biabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqbaoaaCa aaleqajeaibaqcLbmacaaIZaaaaKqbaoaabmaakeaajugibiabeI7a XLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaeq iUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaakiaawIcacaGL PaaajuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRi aaikdacaaI0aGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaigdaaSqa baqcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaju aGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLbsa cqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabgU caRiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaGccaGL OaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaaiAdaaaaakeaaju gibiabgUcaRiaaikdacaaI0aGaeqiUdexcfa4aaSbaaKqaGeaajugW aiaaigdaaSqabaqcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaG OmaaWcbeaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaWgaaqcbasa aKqzadGaaGymaaWcbeaajugibiabgUcaRiabeI7aXLqbaoaaBaaaje aibaqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqa bKqaGeaajugWaiaaiEdaaaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaa qcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqcLbsacqaH4oqC juaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaahaaWcbeqcba saaKqzadGaaGynaaaajugibiabgUcaRiaaikdacaaI0aaakiaawIca caGLPaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaWgaaqcbasaaK qzadGaaGymaaWcbeaajugibiabgUcaRiabeI7aXLqbaoaaBaaajeai baqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabK qaGeaajugWaiaaiIdaaaaaaOGaay5waiaaw2faaaqaaKqbaoaabmaa keaajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaK qbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOm aiaaisdaaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiabeI7aXL qbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqbaoaaCaaaleqajeai baqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaOGaayjkai aawMcaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaBaaajeaibaqc LbmacaaIXaaaleqaaKqzGeGaey4kaSIaeqiUdexcfa4aaSbaaKqaGe aajugWaiaaikdaaSqabaaakiaawIcacaGLPaaajuaGdaahaaWcbeqc basaaKqzadGaaGyoaaaaaaaaaa@7AF0@ .

Estimation of Parameter

Maximum Likelihood Estimate (MLE)

Let ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiEaKqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqz GeGaaiilaiaaykW7caWG4bqcfa4aaSbaaKqaGeaajugWaiaaikdaaS qabaqcLbsacaGGSaGaaGPaVlaadIhajuaGdaWgaaqcbasaaKqzadGa aG4maaWcbeaajugibiaacYcacaaMc8UaaGPaVlaac6cacaGGUaGaai OlaiaaykW7caaMc8UaaiilaiaadIhajuaGdaWgaaqcbasaaKqzadGa amOBaaWcbeaaaOGaayjkaiaawMcaaaaa@57D8@  be a random sample from Rani distribution (1.1). The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb aaaa@3756@ of (1.1) is given by

L= ( θ 5 θ 5 +24 ) n i=1 n ( θ+ x i 4 ) e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0tcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaI1aaaaaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgUcaRiaaikda caaI0aaaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWai aad6gaaaqcfa4aaebCaOqaaKqbaoaabmaakeaajugibiabeI7aXjab gUcaRiaadIhajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGda ahaaWcbeqcbasaaKqzadGaaGinaaaaaOGaayjkaiaawMcaaaWcbaqc LbsacaWGPbGaeyypa0JaaGymaaWcbaqcLbsacaWGUbaacqGHpis1ai aaykW7caWGLbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiaad6ga caaMc8UaeqiUdeNaaGPaVlqadIhagaqeaaaaaaa@6B48@

The natural log likelihood function is thus obtained as

lnL=nln( θ 5 θ 5 +24 )+ i=1 n ln( θ+ x i 4 ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaiaadYeacqGH9aqpcaWGUbGaciiBaiaac6gajuaGdaqadaGc baqcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaiwdaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqc LbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaaaakiaawIcaca GLPaaajugibiabgUcaRKqbaoaaqahakeaajugibiGacYgacaGGUbqc fa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaamiEaKqbaoaaBaaaje aibaqcLbmacaWGPbaaleqaaKqbaoaaCaaaleqajeaibaqcLbmacaaI 0aaaaaGccaGLOaGaayzkaaaaleaajugibiaadMgacqGH9aqpcaaIXa aaleaajugibiaad6gaaiabggHiLdGaeyOeI0IaamOBaiaaykW7cqaH 4oqCcaaMc8UabmiEayaaraaaaa@6C6A@ .
Now dlnL dθ = 5n θ 5n θ 4 θ 5 +24 + i=1 n 1 θ+ x i 4 n x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGUbGaamitaaGcbaqcLbsacaWGKbGa eqiUdehaaiabg2da9KqbaoaalaaakeaajugibiaaiwdacaWGUbaake aajugibiabeI7aXbaacqGHsisljuaGdaWcaaGcbaqcLbsacaaI1aGa aGPaVlaad6gacaaMc8UaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaisdaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbma caaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaaGaey4kaSscfa4aaa bCaOqaaKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaeqiUdeNa ey4kaSIaamiEaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqbao aaCaaaleqajeaibaqcLbmacaaI0aaaaaaaaSqaaKqzGeGaamyAaiab g2da9iaaigdaaSqaaKqzGeGaamOBaaGaeyyeIuoacqGHsislcaWGUb GaaGPaVlqadIhagaqeaaaa@70AB@ , where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean.

The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  is the solution of the equation dlnL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGUbGaamitaaGcbaqcLbsacaWGKbGa eqiUdehaaiabg2da9iaaicdaaaa@3FC3@  and thus it is the solution of the following nonlinear equation

5n θ 5n θ 4 θ 5 +24 + i=1 n 1 θ+ x i 4 n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGynaiaad6gaaOqaaKqzGeGaeqiUdehaaiabgkHiTKqb aoaalaaakeaajugibiaaiwdacaaMc8UaamOBaiaaykW7cqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGinaaaaaOqaaKqzGeGaeqiUdexc fa4aaWbaaSqabKqaGeaajugWaiaaiwdaaaqcLbsacqGHRaWkcaaIYa GaaGinaaaacqGHRaWkjuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGeGa aGymaaGcbaqcLbsacqaH4oqCcqGHRaWkcaWG4bqcfa4aaSbaaKqaGe aajugWaiaadMgaaSqabaqcfa4aaWbaaSqabKqaGeaajugWaiaaisda aaaaaaWcbaqcLbsacaWGPbGaeyypa0JaaGymaaWcbaqcLbsacaWGUb aacqGHris5aiabgkHiTiaad6gacaaMc8UabmiEayaaraGaeyypa0Ja aGimaaaa@6958@

Method of moment Estimate (MOME)

Equating the population mean of Rani distribution (1.1) to the corresponding sample mean, MOME θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaacaaaa@384A@ , of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  is the solution of the following six degree polynomial equation
x ¯ θ 6 θ 5 +24θ x ¯ 120=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacaaMc8UaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiAda aaqcLbsacqGHsislcqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG ynaaaajugibiabgUcaRiaaikdacaaI0aGaaGPaVlabeI7aXjaaykW7 ceWG4bGbaebacqGHsislcaaIXaGaaGOmaiaaicdacqGH9aqpcaaIWa aaaa@515C@ .

A Simulation Study

In this section, a simulation study has been carried out to know the efficiency of the maximum likelihood estimate(MLE) of Rani distribution. The simulation study is based on Acceptance/Rejection method.

Acceptance/Rejection Algorithm:
To simulate from the density f X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaadIfaaSqabaaaaa@3A5F@ , it is assumed that we have envelope density h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb aaaa@3772@  from which it can simulate and that we have some k< MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb GaeyipaWJaeyOhIukaaa@39EA@ such that Sup x f X ( x ) h( x ) k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeaO qaaKqzGeGaae4uaiaabwhacaqGWbaaleaajugibiaadIhaaSqabaqc LbsacaaMc8Ecfa4aaSaaaOqaaKqzGeGaamOzaKqbaoaaBaaajeaiba qcLbmacaWGybaaleqaaKqbaoaabmaakeaajugibiaadIhaaOGaayjk aiaawMcaaaqaaKqzGeGaamiAaKqbaoaabmaakeaajugibiaadIhaaO GaayjkaiaawMcaaaaajugibiabgsMiJkaadUgaaaa@4EF7@ .
Step 1. Simulate X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@  from h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb aaaa@3772@
Step 2. Generate Y~U( 0,kh( x ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb GaaGPaVlaac6hacaWGvbqcfa4aaeWaaOqaaKqzGeGaaGimaiaacYca caWGRbGaaGPaVlaadIgajuaGdaqadaGcbaqcLbsacaWG4baakiaawI cacaGLPaaaaiaawIcacaGLPaaaaaa@4603@ , where k= θ 5 θ 5 +24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqa GeaajugWaiaaiwdaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaGOmaiaaisdaaaaaaa@4646@
Step 3. If Y< f X ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb GaaGPaVlabgYda8iaadAgajuaGdaWgaaqcbasaaKqzadGaamiwaaWc beaajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaaaaa@4183@ , then return X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ , otherwise go to step 1
The simulation study is based on generating N=10,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob Gaeyypa0JaaGymaiaaicdacaGGSaGaaGimaiaaicdacaaIWaaaaa@3CB1@  samples of size n=50,100,150,200 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb Gaeyypa0JaaGynaiaaicdacaGGSaGaaGymaiaaicdacaaIWaGaaiil aiaaigdacaaI1aGaaGimaiaacYcacaaIYaGaaGimaiaaicdaaaa@429A@ for θ=0.5,1,1.5and2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcqGH9aqpcaaIWaGaaiOlaiaaiwdacaGGSaGaaGymaiaacYcacaaI XaGaaiOlaiaaiwdacaaMc8UaaGPaVlaaykW7caqGHbGaaeOBaiaabs gacaaMc8UaaGPaVlaaykW7caaIYaaaaa@4C6D@ using above algorithm. Then we calculate the following measures
(i) Average bias of the simulated estimate

Averagebias= 1 N i=1 N ( θ ^ i θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGbb GaaeODaiaabwgacaqGYbGaaeyyaiaabEgacaqGLbGaaGPaVlaaykW7 caaMc8UaaeOyaiaabMgacaqGHbGaae4CaiaaykW7caaMc8Uaeyypa0 JaaGPaVNqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamOtaaaa juaGdaaeWbGcbaqcfa4aaeWaaOqaaKqzGeGafqiUdeNbaKaajuaGda WgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabgkHiTiabeI7aXbGc caGLOaGaayzkaaaaleaajugibiaadMgacqGH9aqpcaaIXaaaleaaju gibiaad6eaaiabggHiLdaaaa@6050@ , where θ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaaaa@3B4B@  is the ML estimate
(ii) Average mean square error (MSE)

Averagemeansquareerror= 1 N i=1 N ( θ ^ i θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGbb GaaeODaiaabwgacaqGYbGaaeyyaiaabEgacaqGLbGaaGPaVlaaykW7 caaMc8UaaeyBaiaabwgacaqGHbGaaeOBaiaaykW7caaMc8UaaGPaVl aabohacaqGXbGaaeyDaiaabggacaqGYbGaaeyzaiaaykW7caaMc8Ua aeyzaiaabkhacaqGYbGaae4BaiaabkhacaaMc8UaaGPaVlaaykW7cq GH9aqpcaaMc8Ecfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWG obaaaKqbaoaaqahakeaajuaGdaqadaGcbaqcLbsacuaH4oqCgaqcaK qbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGaeyOeI0IaeqiU dehakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaa aaaSqaaKqzGeGaamyAaiabg2da9iaaigdaaSqaaKqzGeGaamOtaaGa eyyeIuoaaaa@76BF@ .

The average bias and average mean square error (MSE) for each of the ML estimate has been calculated and shown in Table (3), where MSE has been shown in bracket.

n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@

θ=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGimaiaac6cacaaI1aaaaa@3B6B@

θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaaaa@39FB@

θ=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aaaaa@3B6C@

θ=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGOmaaaa@39FC@

50

0.05034

0.026212

0.010078

-0.00079

-0.12673

-0.03435

-0.01008

100

0.025405

0.132465

0.005188

-0.00033

-0.06454

-0.01755

-0.00269

150

0.017098

0.008916

0.003523

-0.0002

-0.04385

-0.01193

-0.00186

200

0.012992

0.006755

0.002713

-0.00012

-0.03376

-0.00913

-0.00147

Table 3: Average bias and average mean square error of the simulated estimate.

The graphical presentation of MSE for different values of parameter is shown in Figure 5.

Goodness of Fit

In this section, the goodness of fit of Rani distribution has been discussed with a real lifetime data set from engineering and the fit has been compared with one parameter lifetime distributions namely Akash [3], Shanker [4], Amarendra [7], Aradhana [5], Sujatha [6], Devya [8], Lindley [1] and exponential. The data set is the strength data of glass of the aircraft window reported by Fuller, et al. [14] and are given as 18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.50, 25.52, 25.80, 26.69, 26.77, 26.78, 27.05, 27.67, 29.90, 31.11, 33.20, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, 44.045, 45.29, 45.381. In order to compare lifetime distributions, values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC (Akaike Information Criterion) and K-S Statistic ( Kolmogorov-Smirnov Statistic) for the above data set have been computed and presented in Table (4).

Figure 5: Graphs of MSE for different values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@

The formulae for computing AIC and K-S Statistic are as follows:
AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaamysaiaadoeacqGH9aqpcqGHsislcaaIYaGaciiBaiaac6gacaWG mbGaey4kaSIaaGOmaiaadUgaaaa@40D3@ , K-S= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb GaaeylaiaadofacqGH9aqpjuaGdaWfqaGcbaqcLbsacaqGtbGaaeyD aiaabchaaSqaaKqzGeGaamiEaaWcbeaajuaGdaabdaGcbaqcLbsaca WGgbqcfa4aaSbaaKqaGeaajugWaiaad6gaaSqabaqcfa4aaeWaaOqa aKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGHsislcaWGgbqcfa 4aaSbaaKqaGeaajugWaiaaicdaaSqabaqcfa4aaeWaaOqaaKqzGeGa amiEaaGccaGLOaGaayzkaaaacaGLhWUaayjcSdaaaa@5435@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@  = the number of parameters, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@  = the sample size and F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaad6gaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaaaaa@3E0C@ is the empirical distribution function. The best distribution is the distribution which corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC, and K-S statistic and higher p-value. The MLE ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGafqiUdeNbaKaaaOGaayjkaiaawMcaaaaa@3A76@  with the standard error, S.E ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGafqiUdeNbaKaaaOGaayjkaiaawMcaaaaa@3A76@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ , 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC, K-S Statistic and p-value of the fitted distributions are presented in the Table (4). It can be easily observed from above Table (3) that Rani distribution gives better fit than the fit given by Akash [3], Rama [9], Akshaya [10], Shanker [4], Amarendra [7], Aradhana [5], Sujatha [6], Devya [8], Lindley [1] and exponential distributions and hence it can be considered as an important lifetime distribution for modeling lifetime data over these distributions.

Concluding Remarks

A one parameter lifetime distribution named, “Rani distribution” has been proposed. Its statistical properties including shapes, moments, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves and stress-strength reliability have been discussed. The condition under which Rani distribution is over-dispersed, equi-dispersed, and under-dispersed are presented along other one parameter lifetime distributions. Maximum likelihood estimation and method of moments have been discussed for estimating its parameter. A simulation study has been presented. Finally, the goodness of fit test using K-S Statistic (Kolmogorov-Smirnov Statistic) and p-value for a real lifetime data has been presented and the fit has been compared with some one parameter lifetime distributions.

NOTE: The paper is named “Rani distribution” in the name of my lovely niece Rani Kumari, second daughter of my respected eldest brother Professor Shambhu Sharma, Department of Mathematics, Dayalbagh Educational Institute, Dayalbagh, Agra, India.

Distributions ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacuaH4oqCgaqcaaGaayjkaiaawMcaaaaa@39D3@

( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacuaH4oqCgaqcaaGaayjkaiaawMcaaaaa@39D3@

S.E ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacuaH4oqCgaqcaaGaayjkaiaawMcaaaaa@39D3@

2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@

AIC

K-S

p-value

Rani

0.162278

0.013034

227.25

229.25

0.223

0.0775

Akash

0.097065

0.010048

240.68

242.68

0.298

0.0059

Rama

0.129782

0.011651

232.79

234.79

0.253

0.0301

Akshaya

0.125745

0.011292

234.44

236.44

0.263

0.0223

Shanker

0.647164

0.0082

252.35

254.35

0.358

0.0004

Amarendra

0.128294

0.012413

233.41

235.41

0.257

0.0269

Aradhana

0.094319

0.00978

242.22

244.22

0.306

0.0044

Sujatha

0.095613

0.009904

241.5

243.5

0.303

0.0051

Devya

0.160873

0.012916

227.68

229.68

0.422

0

Lindley

0.062992

0.008001

253.98

255.98

0.365

0.0003

Exponential

0.032449

0.005822

274.52

276.53

0.458

0

Table 4: MLE’s, S.E - 2ln L, AIC and K-S Statistics of the fitted distributions of the given data set

References

  1. Lindley DV (1958) Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society Series B 20: 102-107.
  2. Shanker R, Hagos F (2015) On modeling of Lifetimes data using exponential and Lindley distributions Biometrics & Biostatistics International Journal 2(5): 1-9.
  3. Shanker R (2015 a) Akash distribution and its Applications. International Journal of                        Probability and Statistics 4(3): 65-75.
  4. Shanker R (2015 b) Shanker distribution and its Applications. International Journal of Statistics and Applications5(6): 338-348.
  5. Shanker R (2016 c) Aradhana distribution and its Applications. International Journal of Statistics and Applications 6(1): 23-34.
  6. Shanker R (2016 d) Sujatha distribution and its Applications. Statistics in Transition-new series17(3): 1-20.
  7. Shanker R (2016 a) Amarendra distribution and its Applications. American Journal of Mathematics and Statistics 6(1): 44-56.
  8. Shanker R (2016 b) Devya distribution and its Applications. International Journal of                       Statistics and Applications 6(4): 189-202.
  9. Shanker R (2017 a) Rama distribution and its Application. International Journal of Statistics and Applications 7(1) 26-35.
  10. Shanker R (2017 b) Akshaya distribution and its Application. American Journal of Mathematics and Statistics 7(2) 51-59.
  11. Shaked M and Shanthikumar JG (1994) Stochastic Orders and Their Applications Academic Press, New York, USA.
  12. Bonferroni CE (1930) Elementi di Statistca generale Seeber Firenze.
  13. Renyi A (1961) On measures of entropy and information in proceedings of the 4th berkeley symposium on Mathematical Statistics and Probability Berkeley. Univ of Cali Press 1: 547-561.
  14. Fuller E J, Frieman S, Quinn J, Quinn G, Carter W (1994) Fracture mechanics approach to the design of glass aircraft windows: A case study. SPIE Proc 2286: 419-430.
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