eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 3

Pranav distribution with properties and its applications

Kamlesh Kumar Shukla
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: May 10, 2018 | Published: June 21, 2018

Correspondence: Kamlesh Kumar Shukla, Department of Statistics, College of Science, Eritrea Institute of Technology, Asmara, Eritrea, Email

Citation: Shukla KK. Pranav distribution with properties and its applications. Biom Biostat Int J. 2018;7(3):244‒254. DOI: 10.15406/bbij.2018.07.00215

Abstract

In this paper, a new one parameter life lime distribution has been proposed and named Pranav distribution. Its statistical and mathematical properties have been derived and its stochastic ordering, mean deviations, Bonferroni and Lorenz curves, order statistics, Renyi entropy measure and stress–strength reliability have been discussed. A Simulation study of proposed distribution has also been discussed. For estimating its parameter method of moments and method of maximum likelihood estimation have been discussed. Goodness of fit of proposed distribution has been presented and compared with other lifetime distributions of one parameter.

Keywords: estimation of parameters, mean deviation, moments, order statistics, reliability measures, renyi entropy measure, stochastic ordering, stress–strength reliability

Introduction

Many researchers have been proposed many distributions of one parameter as well as two parameters for modeling life time data. Some of them are giving good results for different data sets from biological and engineering which are considered as life time distribution. But some of them are not giving good results for all the data sets from biological and engineering such as Lindley distribution proposed by Lindley1 Akash and Shanker distributions proposed by Shanker2,3 and Ishita distribution proposed by Shanker.4 These distributions are giving good fit over exponential distribution whereas Akash distribution proposed by Shanker2 has been applied on biological data and mentioned the superiority over Lindley distribution and Exponential distribution.

The detailed study about its mathematical properties, estimation of parameter and application has been shown in their paper. Sujatha distribution proposed by Shanker5 has also been applied on real life time data from biological and engineering. Its mathematical and statistical properties has also been discussed in that paper and showed on some selected data set, better than Lindley and Exponential distribution.

Ghitany6 reported in their paper that Lindley distribution is superior to exponential distribution with reference to data related the waiting time before service of the bank customers. In this regard, author interest was to propose new life time distribution which may give good fit in compare to other life time distribution. That is main motivation to propose a new life distribution and applied on biological data.

Therefore, searching a new life time distribution which may be better than Lindley, Exponential, Ishita, Shanker, Sujatha and Akash distribution. It is describe as below:

One parameter life time distribution having parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ is defined by its pdf

f( x;θ )= θ 4 θ 4 +6 ( θ+ x 3 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaeaajugibiaadIhacaGG7aGaeqiUdehajuaGcaGLOaGa ayzkaaqcLbsacqGH9aqpjuaGdaWcaaqaaKqzGeGaeqiUdexcfa4aaW baaeqajuaibaqcLbmacaaI0aaaaaqcfayaaKqzGeGaeqiUdehddaah aaqcfasabeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaaaKqbao aabmaabaqcLbsacqaH4oqCcqGHRaWkcaWG4baddaahaaqcfasabeaa jugWaiaaiodaaaaajuaGcaGLOaGaayzkaaqcLbsacaWGLbaddaahaa qcfasabeaajugWaiabgkHiTiabeI7aXjaadIhaaaqcLbsacaGG7aGa amiEaiabg6da+iaaicdacaGGSaGaeqiUdeNaeyOpa4JaaGimaaaa@65A1@ (1.1)

The pdf (1.1) would call ‘Pranav distribution’ which is a mixture of two–distributions , exponential distribution having scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and gamma distribution having shape parameter 4 and scale parameter,  and their mixing proportions of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ θ 4 ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaajuaGba qcLbsacqaH4oqCkmaaCaaajuaGbeqcfasaaKqzadGaaGinaaaaaKqb agaakmaabmaajuaGbaqcLbsacqaH4oqCkmaaCaaajuaGbeqcfasaaK qzadGaaGinaaaajugibiabgUcaRiaaiAdaaKqbakaawIcacaGLPaaa aaaaaa@4868@ and 6 ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba qcLbsacaaI2aaajuaGbaWaaeWabeaajugibiabeI7aXLqbaoaaCaaa beqcKvaq=haajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaajuaGca GLOaGaayzkaaaaaaaa@4341@ respectively.

f 2 ( x;θ )=p g 1 ( x;θ )+(1p) g 2 ( x;θ,4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0R1xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb GcdaWgaaqcfasaaKqzGeGaaGOmaaqcfayabaGcdaqadaqcfayaaKqz GeGaamiEaiaacUdacqaH4oqCaKqbakaawIcacaGLPaaajugibiabg2 da9iaadchacaWGNbGcdaWgaaqcfasaaKqzGeGaaGymaaqcfayabaGc daqadaqcfayaaKqzGeGaamiEaiaacUdacqaH4oqCaKqbakaawIcaca GLPaaajugibiabgUcaRiaacIcacaaIXaGaeyOeI0IaamiCaiaacMca caWGNbGcdaWgaaqcfasaaKqzGeGaaGOmaaqcfasabaGcdaqadaqcfa yaaKqzGeGaamiEaiaacUdacqaH4oqCcaGGSaGaaGinaaqcfaOaayjk aiaawMcaaaaa@5D9E@

Where p= θ 4 ( θ 4 +6 ) , g 1 ( x )=θ e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiCai abg2da9KqbaoaalaaakeaajugibiabeI7aXLqbaoaaCaaajeaibeqc Kfay=haajugWaiaaisdaaaaakeaajuaGdaqadaGcbaqcLbsacqaH4o qCjuaGdaahaaqcbasabKazba2=baqcLbmacaaI0aaaaKqzGeGaey4k aSIaaGOnaaGccaGLOaGaayzkaaaaaKqzGeGaaiilaiaadEgajuaGda WgaaqcbasaaKqzGeGaaGymaaqcbasabaqcfa4aaeWaaOqaaKqzGeGa amiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcqaH4oqCcaGGLbqcfa 4aaWbaaSqabKazba2=baqcLbmacqGHsislcqaH4oqCcaWG4baaaaaa @5EEC@ and g 2 ( x )= θ 4 x 3 e θx 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaK qbaoaaBaaajqwaG9FaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWG4baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiabeI7aXLqbaoaaCaaajeaibeqcKfay=haajugWaiaaisda aaqcLbsacaWG4bqcfa4aaWbaaKqaGeqajqwaG9FaaKqzadGaaG4maa aajugibiaacwgajuaGdaahaaWcbeqcKfay=haajugWaiabgkHiTiab eI7aXjaadIhaaaaakeaajugibiaaiAdaaaaaaa@58E8@

The corresponding cumulative distribution function (cdf) of (1.1) is given by

F( x;θ )=1[ 1+ θx( θ 2 x 2 +3θx+6 ) θ 4 +6 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaaGymaiabgkHiTKqbaoaadmaakeaajugibi aaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacqaH4oqCcaWG4bqcfa4a aeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcLbsacaWG4bWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsa cqGHRaWkcaaIZaGaeqiUdeNaamiEaiabgUcaRiaaiAdaaOGaayjkai aawMcaaaqaaKqzGeGaeqiUde3cdaahaaqcbasabeaajugWaiaaisda aaqcLbsacqGHRaWkcaaI2aaaaaGccaGLBbGaayzxaaqcLbsacaWGLb qcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabeI7aXjaadIhaaaqc LbsacaGG7aGaamiEaiabg6da+iaaicdacaGGSaGaeqiUdeNaeyOpa4 JaaGimaaaa@710C@ (1.2)

In this study, new one parameter life time distribution has been proposed and named Pranav distribution. Moment and its related measures have been discussed. Its hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, order statistics, Renyi entropy measure and stress–strength reliability have been discussed. A simulation study of proposed distribution has also been discussed. Both the method of moments and the method of maximum likelihood estimation have been discussed for estimating the parameter of Pranav distribution. The goodness of fit of the Pranav distribution has been presented and the fit has also been compared with other lifetime distributions of one parameter.

Graphs of the pdf and the cdf of Pranav distribution for varying values of parameter are presented in Figure 1 & 2.

Figure 1 Pdf plots of Pranav distribution of varying values of parameter θ.
Figure 2 Cdf plots of Pranav distribution of varying values of parameter θ.

Moment and its related measure

The r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th moment about origin of Pranav distribution can be obtained as

μ r = r![ θ 4 +( r+1 )( r+2 ) ]( r+3 ) θ r ( θ 4 +6 ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaWGYbaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi aadkhacaGGHaqcfa4aamWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqa bKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcLb sacaWGYbGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOqa aKqzGeGaamOCaiabgUcaRiaaikdaaOGaayjkaiaawMcaaaGaay5wai aaw2faaKqbaoaabmaakeaajugibiaadkhacqGHRaWkcaaIZaaakiaa wIcacaGLPaaaaeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmaca WGYbaaaKqbaoaabmaakeaajugibiabeI7aXTWaaWbaaKqaGeqabaqc LbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaaGccaGLOaGaayzkaaaaaK qzGeGaai4oaiaadkhacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGa aG4maiaacYcacaGGUaGaaiOlaiaac6caaaa@7461@ (2.1)

Thus the first four moments about origin of Pranav distribution are given by

μ 1 = θ 4 +24 θ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaIXaaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4k aSIaaGOmaiaaisdaaOqaaKqzGeGaeqiUdexcfa4aaeWaaOqaaKqzGe GaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGH RaWkcaaI2aaakiaawIcacaGLPaaaaaaaaa@5510@ , μ 2 = 2( θ 4 +60) θ 2 ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaIYaaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi aaikdacaGGOaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisda aaqcLbsacqGHRaWkcaaI2aGaaGimaiaacMcaaOqaaKqzGeGaeqiUde xcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcfa4aaeWaaOqaaKqz GeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacq GHRaWkcaaI2aaakiaawIcacaGLPaaaaaaaaa@59F5@ ,

μ 3 = 6( θ 4 +120) θ 3 ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaIZaaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi aaiAdacaGGOaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisda aaqcLbsacqGHRaWkcaaIXaGaaGOmaiaaicdacaGGPaaakeaajugibi abeI7aXLqbaoaaCaaaleqabaqcLbsacaaIZaaaaKqbaoaabmaakeaa jugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGe Gaey4kaSIaaGOnaaGccaGLOaGaayzkaaaaaaaa@59E9@ , μ 4 = 24( θ 4 +210) θ 4 ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaI0aaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi aaikdacaaI0aGaaiikaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbma caaI0aaaaKqzGeGaey4kaSIaaGOmaiaaigdacaaIWaGaaiykaaGcba qcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajuaG daqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG inaaaajugibiabgUcaRiaaiAdaaOGaayjkaiaawMcaaaaaaaa@5B6E@

Using above raw moments about origin, central moments of Pranav distribution are thus derived as

μ 2 = ( θ 8 +84 θ 4 +144) θ 2 ( θ 4 +6 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiaacIcacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGioaaaajugibiabgUcaRiaaiIdacaaI0aGaeqiUdexcfa4a aWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaIXaGaaG inaiaaisdacaGGPaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaIYaaaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCa aaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaaGccaGL OaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaaaaaaa@6038@

μ 3 = 2( θ 12 +198 θ 8 +324 θ 4 +864) θ 3 ( θ 4 +6 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaG4maaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiaaikdacaGGOaGaeqiUdexcfa4aaWbaaSqabK qaGeaajugWaiaaigdacaaIYaaaaKqzGeGaey4kaSIaaGymaiaaiMda caaI4aGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiIdaaaqcLb sacqGHRaWkcaaIZaGaaGOmaiaaisdacqaH4oqCjuaGdaahaaWcbeqc basaaKqzadGaaGinaaaajugibiabgUcaRiaaiIdacaaI2aGaaGinai aacMcaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaa iodaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGe aajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaakiaawIcacaGLPaaa juaGdaahaaWcbeqcbasaaKqzadGaaG4maaaaaaaaaa@6AA8@

μ 4 = 9( θ 16 +312 θ 12 +2304 θ 8 +10368 θ 4 +10368) θ 4 ( θ 4 +6 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGinaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiaaiMdacaGGOaGaeqiUdexcfa4aaWbaaSqabK azba4=baqcLbkacaaIXaGaaGOnaaaajugibiabgUcaRiaaiodacaaI XaGaaGOmaiabeI7aXLqbaoaaCaaaleqajqwaa+FaaKqzGcGaaGymai aaikdaaaqcLbsacqGHRaWkcaaIYaGaaG4maiaaicdacaaI0aGaeqiU dexcfa4aaWbaaSqabKqaGeaajugWaiaaiIdaaaqcLbsacqGHRaWkca aIXaGaaGimaiaaiodacaaI2aGaaGioaiabeI7aXLqbaoaaCaaaleqa jeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGymaiaaicdacaaIZa GaaGOnaiaaiIdacaGGPaaakeaajugibiabeI7aXLqbaoaaCaaaleqa jeaibaqcLbmacaaI0aaaaKqbaoaabmaakeaajugibiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaaGc caGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaaaaa aa@7A83@

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 qaaKqbaoaakaaakeaajugibiabek7aILqbaoaaBaaajqwaa+FaaKqz GeGaaGymaaWcbeaaaeqaaaGccaGLOaGaayzkaaaaaa@3EEA@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHYoGylmaaBaaabaqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaaa aa@3B4B@ and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeq4SdCgakiaawIcacaGLPaaaaaa@3A57@ of Pranav distribution are obtained as

C.V= σ μ 1 = ( θ 8 +84 θ 4 +144) ( θ 4 +24 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb GaaiOlaiaadAfacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCaOqa aKqzGeGaeqiVd02cdaWgaaqcbasaaKqzadGaaGymaaqcbasabaWcda ahaaqcbasabeaajugWaiadacUHYaIOaaaaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqbaoaakaaakeaajugibiaacIcacqaH4oqCjuaGdaahaa WcbeqcbasaaKqzadGaaGioaaaajugibiabgUcaRiaaiIdacaaI0aGa eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRa WkcaaIXaGaaGinaiaaisdacaGGPaaaleqaaaGcbaqcfa4aaeWabOqa aKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLb sacqGHRaWkcaaIYaGaaGinaaGccaGLOaGaayzkaaaaaaaa@64B9@

β 1 = μ 3 μ 2 3/2 = 2( θ 12 +198 θ 8 +324 θ 4 +864) ( θ 8 +84 θ 4 +144 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaa beaajugibiabg2da9KqbaoaalaaakeaajugibiabeY7aTTWaaSbaaK qaGeaajugWaiaaiodaaKqaGeqaaaGcbaqcLbsacqaH8oqBjuaGdaWg aaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaahaaWcbeqcbasaaKqzad GaaG4maiaac+cacaaIYaaaaaaajugibiabg2da9Kqbaoaalaaakeaa jugibiaaikdacaGGOaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaigdacaaIYaaaaKqzGeGaey4kaSIaaGymaiaaiMdacaaI4aGaeqiU dexcfa4aaWbaaSqabKqaGeaajugWaiaaiIdaaaqcLbsacqGHRaWkca aIZaGaaGOmaiaaisdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGa aGinaaaajugibiabgUcaRiaaiIdacaaI2aGaaGinaiaacMcaaOqaaK qbaoaabmGakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbma caaI4aaaaKqzGeGaey4kaSIaaGioaiaaisdacqaH4oqCjuaGdaahaa WcbeqcbasaaKqzadGaaGinaaaajugibiabgUcaRiaaigdacaaI0aGa aGinaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaaio dacaGGVaGaaGOmaaaaaaaaaa@80D1@

β 2 = μ 4 μ 2 2 = 9( θ 16 +312 θ 12 +2304 θ 8 +10368 θ 4 +10368) ( θ 8 +84 θ 4 +144 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiabeY7aTTWaaSbaaKqaGeaajugWaiaaisdaaK qaGeqaaaGcbaqcLbsacqaH8oqBmmaaBaaajqwaa+FaaKqzGcGaaGOm aaqcKfaG=hqaaWWaaWbaaKazba4=beqaaKqzGcGaaGOmaaaaaaqcLb sacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaI5aGaaiikaiabeI7aXLqb aoaaCaaajqwaa+FabeaajugOaiaaigdacaaI2aaaaKqzGeGaey4kaS IaaG4maiaaigdacaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaigdacaaIYaaaaKqzGeGaey4kaSIaaGOmaiaaiodacaaIWaGaaG inaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI4aaaaKqzGeGa ey4kaSIaaGymaiaaicdacaaIZaGaaGOnaiaaiIdacqaH4oqCjuaGda ahaaqcbasabeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaIXaGaaGim aiaaiodacaaI2aGaaGioaiaacMcaaOqaaKqbaoaabmaakeaajugibi abeI7aXLqbaoaaCaaajeaibeqaaKqzadGaaGioaaaajugibiabgUca RiaaiIdacaaI0aGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaais daaaqcLbsacqGHRaWkcaaIXaGaaGinaiaaisdaaOGaayjkaiaawMca aKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaaaaaa@8F76@

γ= σ 2 μ 1 = ( θ 8 +84 θ 4 +144) θ( θ 4 +6)( θ 4 +24) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWClmaaCaaajeaibeqa aKqzadGaaGOmaaaaaOqaaKqzGeGaeqiVd02cdaWgaaqcbasaaKqzad GaaGymaaqcbasabaWcdaahaaqcbasabeaajugWaiadacUHYaIOaaaa aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaiikaiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaaI4aaaaKqzGeGaey4kaSIaaGioaiaa isdacqaH4oqCmmaaCaaajqwaa+FabeaajugOaiaaisdaaaqcLbsacq GHRaWkcaaIXaGaaGinaiaaisdacaGGPaaakeaajugibiabeI7aXjaa cIcacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibi abgUcaRiaaiAdacaGGPaGaaiikaiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOmaiaaisdacaGGPaaaaa aa@6FA3@

To study the nature of C.V, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdi2cdaWgaaqcKfaG=haajugWaiaaigdaaKazba4= beaaaSqabaaaaa@3EC8@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@3AF4@ , and of Pranav distribution, graphs of C.V, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdi2cdaWgaaqcbasaaKqzadGaaGymaaqcbasabaaa leqaaaaa@3B42@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@3AF4@ , and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo Wzaaa@382C@ of Pranav distribution have been drawn for varying values of the parameters and presented in Figure 3.

Figure 3 CV, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdi2cdaWgaaqcbasaaKqzadGaaGymaaqcbasabaaa leqaaaaa@3B42@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@3AF4@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo Wzaaa@382C@ of Pranav distribution.

Reliability measures

Let X be a continuous random variable with pdf f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B27@ and cdf F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@3B07@ . The hazard rate function (also known as the failure rate function) and the mean residual life function of are respectively defined as

h( x )= f( x ) 1F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaWGMbqcfa4aaeWaaOqaaKqzGeGaam iEaaGccaGLOaGaayzkaaaabaqcLbsacaaIXaGaeyOeI0IaamOraKqb aoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaaaaaa@4950@ (3.1)

 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 4baakiaawUfacaGLDbaajugibiabg2da9Kqbaoaalaaakeaajugibi aaigdaaOqaaKqzGeGaaGymaiabgkHiTiaadAeajuaGdaqadaGcbaqc LbsacaWG4baakiaawIcacaGLPaaaaaqcfa4aa8qmaOqaaKqbaoaadm aakeaajugibiaaigdacqGHsislcaWGgbqcfa4aaeWaaOqaaKqzGeGa amiDaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaajeaibaqcLbmaca WG4baajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipacaWGKbGaamiD aaaa@6784@ (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 Pranav distribution (1.1) are obtained as

h( x )= θ 4 ( θ+ x 3 ) ( θ 3 x 3 +3 θ 2 x 2 +6θx+ θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqClmaaCaaajeaibeqaaKqzad GaaGinaaaajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaWG4bqc fa4aaWbaaSqabKqaGeaajugWaiaaiodaaaaakiaawIcacaGLPaaaae aajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaG4maaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaG 4maaaajugibiabgUcaRiaaiodacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzad GaaGOmaaaajugibiabgUcaRiaaiAdacqaH4oqCcaWG4bGaey4kaSIa eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRa WkcaaI2aaakiaawIcacaGLPaaaaaaaaa@6E35@ (3.3)

and m( x )= 1 ( θ 3 x 3 +3 θ 2 x 2 +6θx+ θ 4 +6 ) x ( θ 3 t 3 +3 θ 2 t 2 +6θt+ θ 4 +6 ) e θt dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajuaGdaqadaGcbaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiaa dIhajuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabgUcaRi aaiodacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugi biaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgU caRiaaiAdacqaH4oqCcaWG4bGaey4kaSIaeqiUdexcfa4aaWbaaSqa bKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaakiaawIcaca GLPaaaaaqcfa4aa8qCaOqaaKqbaoaabmaakeaajugibiabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaamiDaKqbaoaaCa aaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaG4maiabeI7a XLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaamiDaKqbao aaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOnaiab eI7aXjaadshacqGHRaWkcqaH4oqCjuaGdaahaaWcbeqcbasaaKqzad GaaGinaaaajugibiabgUcaRiaaiAdaaOGaayjkaiaawMcaaKqzGeGa amyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH4oqCcaWG0b aaaKqzGeGaamizaiaadshaaKqaGeaajugWaiaadIhaaKqaGeaajugW aiabg6HiLcqcLbsacqGHRiI8aaaa@9732@

= ( θ 3 x 3 +6 θ 2 x 2 +18θx+ θ 4 +24 ) θ( θ 3 x 3 +3 θ 2 x 2 +6θx+ θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWba aSqabKqaGeaajugWaiaaiodaaaqcLbsacaWG4bqcfa4aaWbaaSqabK qaGeaajugWaiaaiodaaaqcLbsacqGHRaWkcaaI2aGaeqiUdexcfa4a aWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacaWG4bqcfa4aaWbaaS qabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIXaGaaGioaiab eI7aXjaadIhacqGHRaWkcqaH4oqCjuaGdaahaaWcbeqcbasaaKqzad GaaGinaaaajugibiabgUcaRiaaikdacaaI0aaakiaawIcacaGLPaaa aeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeI7aXLqbaoaaCa aaleqajeaibaqcLbmacaaIZaaaaKqzGeGaamiEaKqbaoaaCaaaleqa jeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaG4maiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaamiEaKqbaoaaCaaa leqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOnaiabeI7aXj aadIhacqGHRaWkcqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGin aaaajugibiabgUcaRiaaiAdaaOGaayjkaiaawMcaaaaaaaa@8242@ (3.4)

 It can be easily verified that h( 0 )= θ 5 θ 4 +6 =f( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaaGimaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGynaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaaaiabg2da9iaadAgaju aGdaqadaGcbaqcLbsacaaIWaaakiaawIcacaGLPaaaaaa@4EF0@ and m( 0 )= θ 4 +24 θ( θ 4 +6 ) = μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaaGimaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGinaaaajugibiabgUcaRiaaikdacaaI0aaakeaajugibiab eI7aXLqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaaGccaGLOaGaayzkaaaa aKqzGeGaeyypa0JaeqiVd02cdaWgaaqcbasaaKqzadGaaGymaaqcba sabaWcdaahaaqcbasabeaajugWaiadacUHYaIOaaaaaa@5B0B@ .

The graph 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@ of Pranav distribution for varying values of parameter is shown in Figure 4&5 respectively.

Figure 4 h(x) of Pranav distribution for varying value of parameter θ.
Figure 5 m(x) of Pranav distribution for varying values of parameter θ.

Stochastic orderings

Stochastic ordering of positive continuous random variables is crucial method for evaluating their comparative behavior. A random variable X is said to be smaller than a random variable Y in the

stochastic order ( X st Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiwaiabgsMiJUWaaSbaaKqaGeaajugWaiaadohacaWG 0baajeaibeaajugibiaadMfaaOGaayjkaiaawMcaaaaa@404E@ 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@

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

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

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.

 

The following results due to Shaked7 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 GaeyizImAcfa4aaSbaaKqaGeaajugWaiaadYgacaWGYbaaleqaaKqz GeGaamywaiabgkDiElaadIfacqGHKjYOjuaGdaWgaaqcbasaaKqzad GaamiAaiaadkhaaSqabaqcLbsacaWGzbGaeyO0H4TaamiwaiabgsMi JMqbaoaaBaaajeaibaqcLbmacaWGTbGaamOCaiaadYgaaSqabaqcLb sacaWGzbaaaa@5418@

X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeaK aaGeaajugWaiabgoDiFdWcbaqcLbsacaWGybGaeyizImAcfa4aaSba aKGaGeaajugWaiaadohacaWG0baameqaaKqzGeGaamywaaWcbeaaaa a@4311@

The Pranav distribution is ordered with respect to the strongest ‘likelihood ratio ordering’ as established in the following theorem:

Theorem: Let X and Y follows Pranav distribution with parameters ( θ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaa kiaawIcacaGLPaaaaaa@3D33@ and ( θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaa kiaawIcacaGLPaaaaaa@3D34@ respectively. If θ 1 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabgwMiZkab eI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaaa@41E1@ 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; θ 1 ) f Y ( x; θ 2 ) = θ 1 4 ( θ 2 4 +6 ) θ 2 4 ( θ 1 4 +6 ) e ( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSqaaS qaaKqzGeGaamOzaKqbaoaaBaaajiaibaqcLbmacaWGybaameqaaKqb aoaabmaaleaajugibiaadIhacaGG7aGaeqiUde3cdaWgaaadbaqcLb macaaIXaaameqaaaWccaGLOaGaayzkaaaabaqcLbsacaWGMbqcfa4a aSbaaKGaGeaajugWaiaadMfaaWqabaqcfa4aaeWaaSqaaKqzGeGaam iEaiaacUdacqaH4oqClmaaBaaajiaibaqcLbmacaaIYaaajiaibeaa aSGaayjkaiaawMcaaaaajugibiabg2da9Kqbaoaalaaakeaajugibi abeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaajeaibeaajuaGdaah aaqcbasabeaajugWaiaaisdaaaqcfa4aaeWaaOqaaKqzGeGaeqiUde xcfa4aaSbaaKqaGeaajugWaiaaikdaaKqaGeqaaKqbaoaaCaaajeai beqaaKqzadGaaGinaaaajugibiabgUcaRiaaiAdaaOGaayjkaiaawM caaaqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaKqa GeqaaKqbaoaaCaaajeaibeqaaKqzadGaaGinaaaajuaGdaqadaGcba qcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaqcbasabaqc fa4aaWbaaKqaGeqabaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaa GccaGLOaGaayzkaaaaaKqzGeGaamyzaKqbaoaaCaaajeaibeqaaKqz adGaeyOeI0scfa4aaeWaaKqaGeaajugWaiabeI7aXLqbaoaaBaaaji aibaqcLbmacaaIXaaajiaibeaajugWaiabgkHiTiabeI7aXLqbaoaa BaaajiaibaqcLbmacaaIYaaajiaibeaaaKqaGiaawIcacaGLPaaaju gWaiaadIhaaaaaaa@90D5@ ; x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGG7a GaaeiiaiaadIhacqGH+aGpcaaIWaaaaa@3AA6@

Now

ln f X ( x; θ 1 ) f Y ( x; θ 2 ) =ln[ θ 1 4 ( θ 2 4 +6 ) θ 2 4 ( θ 1 4 +6 ) ]( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaKqbaoaaleaaleaajugibiaadAgajuaGdaWgaaqccasaaKqz adGaamiwaaadbeaajuaGdaqadaWcbaqcLbsacaWG4bGaai4oaiabeI 7aXTWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaWccaGLOaGaayzk aaaabaqcLbsacaWGMbqcfa4aaSbaaKGaGeaajugWaiaadMfaaWqaba qcfa4aaeWaaSqaaKqzGeGaamiEaiaacUdacqaH4oqCjuaGdaWgaaqc casaaKqzadGaaGOmaaadbeaaaSGaayjkaiaawMcaaaaajugibiabg2 da9iGacYgacaGGUbqcfa4aamWaaOqaaKqbaoaalaaakeaajugibiab eI7aXTWaaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaSWaaWbaaKqaGe qabaqcLbmacaaI0aaaaKqbaoaabmaakeaajugibiabeI7aXTWaaSba aKqaGeaajugWaiaaikdaaKqaGeqaaSWaaWbaaKqaGeqabaqcLbmaca aI0aaaaKqzGeGaey4kaSIaaGOnaaGccaGLOaGaayzkaaaabaqcLbsa cqaH4oqClmaaBaaajeaibaqcLbmacaaIYaaajeaibeaalmaaCaaaje aibeqaaKqzadGaaGinaaaajuaGdaqadaGcbaqcLbsacqaH4oqClmaa BaaajeaibaqcLbmacaaIXaaajeaibeaalmaaCaaajeaibeqaaKqzad GaaGinaaaajugibiabgUcaRiaaiAdaaOGaayjkaiaawMcaaaaaaiaa wUfacaGLDbaajugibiabgkHiTKqbaoaabmaakeaajugibiabeI7aXT WaaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaKqzGeGaeyOeI0IaeqiU dexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaakiaawIcacaGLPa aajugibiaadIhaaaa@8E8B@

This gives d dx ln f X ( x; θ 1 ) f Y ( x; θ 2 ) =( θ 1 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaaGcbaqcLbsacaWGKbGaamiEaaaaciGGSbGaaiOB aKqbaoaaleaaleaajugibiaadAgajuaGdaWgaaqccasaaKqzadGaam iwaaadbeaajuaGdaqadaWcbaqcLbsacaWG4bGaai4oaiabeI7aXTWa aSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaWccaGLOaGaayzkaaaaba qcLbsacaWGMbWcdaWgaaqccasaaKqzadGaamywaaqccasabaqcfa4a aeWaaSqaaKqzGeGaamiEaiaacUdacqaH4oqCjuaGdaWgaaqccasaaK qzadGaaGOmaaadbeaaaSGaayjkaiaawMcaaaaajugibiabg2da9iab gkHiTKqbaoaabmaakeaajugibiabeI7aXTWaaSbaaKqaGeaajugWai aaigdaaKqaGeqaaKqzGeGaeyOeI0IaeqiUdexcfa4aaSbaaKqaGeaa jugWaiaaikdaaSqabaaakiaawIcacaGLPaaaaaa@6688@

 Thus if θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qClmaaBaaajeaibaqcLbmacaaIXaaajeaibeaajugibiabg6da+iab eI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaaa@40BF@ or θ 1 = θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qClmaaBaaajeaibaqcLbmacaaIXaaajeaibeaajugibiabg2da9iab eI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaaa@40BD@ , d dx ln f X ( x; θ 1 ) f Y ( x; θ 2 ) <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaaGcbaqcLbsacaWGKbGaamiEaaaaciGGSbGaaiOB aKqbaoaaleaaleaajugibiaadAgajuaGdaWgaaqccasaaKqzadGaam iwaaadbeaajuaGdaqadaWcbaqcLbsacaWG4bGaai4oaiabeI7aXTWa aSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaWccaGLOaGaayzkaaaaba qcLbsacaWGMbWcdaWgaaqccasaaKqzadGaamywaaqccasabaqcfa4a aeWaaSqaaKqzGeGaamiEaiaacUdacqaH4oqClmaaBaaajiaibaqcLb macaaIYaaajiaibeaaaSGaayjkaiaawMcaaaaajugibiabgYda8iaa icdaaaa@5916@ . 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 mean deviation about the mean and median are defined by

δ 1 ( X )= 0 | xμ | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehake aajuaGdaabdaGcbaqcLbsacaWG4bGaeyOeI0IaeqiVd0gakiaawEa7 caGLiWoaaKqaGeaajugWaiaaicdaaKqaGeaajugWaiabg6HiLcqcLb sacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIca caGLPaaajugibiaadsgacaWG4baaaa@5780@ 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 ayjcSdaajeaibaqcLbmacaaIWaaajeaibaqcLbmacqGHEisPaKqzGe Gaey4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGa ayzkaaqcLbsacaWGKbGaamiEaaaa@569D@ , 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 aKqbaoaabmaakeaajugibiaadIfaaOGaayjkaiaawMcaaaaa@41F7@ . The measures δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azlmaaBaaajeaibaqcLbmacaaIXaaajeaibeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaaaaa@3E2A@ and δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbauaaKqzGdGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaaaaa@3ECF@ can be calculated using the following 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 caGLPaaaaKqaGeaajugWaiaaicdaaKqaGeaajugWaiabeY7aTbqcLb sacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIca caGLPaaajugibiaadsgacaWG4bGaey4kaSscfa4aa8qCaOqaaKqbao aabmaakeaajugibiaadIhacqGHsislcqaH8oqBaOGaayjkaiaawMca aaqcbasaaKqzadGaeqiVd0gajeaibaqcLbmacqGHEisPaKqzGeGaey 4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzk aaqcLbsacaWGKbGaamiEaaaa@6DCE@

=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 caGLPaaajugibiabgkHiTKqbaoaapehakeaajugibiaadIhacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacaWG KbGaamiEaaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaeqiVd0gaju gibiabgUIiYdGaeyOeI0IaeqiVd0wcfa4aamWaaOqaaKqzGeGaaGym aiabgkHiTiaadAeajuaGdaqadaGcbaqcLbsacqaH8oqBaOGaayjkai aawMcaaaGaay5waiaaw2faaKqzGeGaey4kaSscfa4aa8qCaOqaaKqz GeGaamiEaaqcbasaaKqzadGaeqiVd0gajeaibaqcLbmacqGHEisPaK qzGeGaey4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGL OaGaayzkaaqcLbsacaWGKbGaamiEaaaa@7135@

=2μF( μ )2μ+2 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIYaGaeqiVd0MaamOraKqbaoaabmaakeaajugibiabeY7aTbGc caGLOaGaayzkaaqcLbsacqGHsislcaaIYaGaeqiVd0Maey4kaSIaaG OmaKqbaoaapehakeaajugibiaadIhaaKqaGeaajugWaiabeY7aTbqc basaaKqzadGaeyOhIukajugibiabgUIiYdGaamOzaKqbaoaabmaake aajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaamizaiaadIhaaaa@56B6@

=2μF( μ )2 0 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIYaGaeqiVd0MaamOraKqbaoaabmaakeaajugibiabeY7aTbGc caGLOaGaayzkaaqcLbsacqGHsislcaaIYaqcfa4aa8qCaOqaaKqzGe GaamiEaaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaeqiVd0gajugi biabgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkai aawMcaaKqzGeGaamizaiaadIhaaaa@52AB@ (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 ayzkaaaajeaqbaqcLboacaaIWaaajeaibaqcLbmacaWGnbaajugibi abgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaa wMcaaKqzGeGaamizaiaadIhacqGHRaWkjuaGdaWdXbGcbaqcfa4aae WaaOqaaKqzGeGaamiEaiabgkHiTiaad2eaaOGaayjkaiaawMcaaaqc basaaKqzadGaamytaaqcbasaaKqzadGaeyOhIukajugibiabgUIiYd GaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhaaaa@6A7F@

=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 qpcaWGnbGaamOraKqbaoaabmaakeaajugibiaad2eaaOGaayjkaiaa wMcaaKqzGeGaeyOeI0scfa4aa8qCaOqaaKqzGeGaamiEaiaadAgaju aGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaajugibiaadsga caWG4baajeaqbaqcLboacaaIWaaajeaibaqcLbmacaWGnbaajugibi abgUIiYdGaeyOeI0IaamytaKqbaoaadmaakeaajugibiaaigdacqGH sislcaWGgbqcfa4aaeWaaOqaaKqzGeGaamytaaGccaGLOaGaayzkaa aacaGLBbGaayzxaaqcLbsacqGHRaWkjuaGdaWdXbGcbaqcLbsacaWG 4baajeaibaqcLbmacaWGnbaajeaibaqcLbmacqGHEisPaKqzGeGaey 4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzk aaqcLbsacaWGKbGaamiEaaaa@6C1D@

=μ+2 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqGHsislcqaH8oqBcqGHRaWkcaaIYaqcfa4aa8qCaOqaaKqzGeGa amiEaaqcbasaaKqzadGaamytaaqcbasaaKqzadGaeyOhIukajugibi abgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaa wMcaaKqzGeGaamizaiaadIhaaaa@4CDA@

=μ2 0 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqaH8oqBcqGHsislcaaIYaqcfa4aa8qCaOqaaKqzGeGaamiEaaqc basaaKqzadGaaGimaaqcbasaaKqzadGaamytaaqcLbsacqGHRiI8ai aadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaajugi biaadsgacaWG4baaaa@4B41@ (5.2)

Using pdf (1.1) and the mean of Pranav distribution, it can be written as:

0 μ x f( x;θ )dx=μ { θ 5 μ+( μ 4 +1 ) θ 4 +4 θ 3 μ 3 +12 θ 2 μ 2 +24θμ+24 } e θμ θ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaaqcbauaaKqzGdGaaGimaaqcbasaaKqzadGaeqiV d0gajugibiabgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaca GG7aGaeqiUdehakiaawIcacaGLPaaajugibiaadsgacaWG4bGaeyyp a0JaeqiVd0MaeyOeI0scfa4aaSaaaOqaaKqbaoaacmaakeaajugibi abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaeqiV d0Maey4kaSscfa4aaeWaaOqaaKqzGeGaeqiVd02cdaahaaqcbasabe aajugWaiaaisdaaaqcLbsacqGHRaWkcaaIXaaakiaawIcacaGLPaaa jugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGe Gaey4kaSIaaGinaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI ZaaaaKqzGeGaeqiVd0wcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaa qcLbsacqGHRaWkcaaIXaGaaGOmaiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacGaAaIOmaaaajugibiabeY7aTLqbaoaaCaaaleqajeaiba qcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaiaaisdacqaH4oqCcqaH 8oqBcqGHRaWkcaaIYaGaaGinaaGccaGL7bGaayzFaaqcLbsacaWGLb qcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabeI7aXjabeY7aTbaa aOqaaKqzGeGaeqiUdexcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaW baaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaakiaa wIcacaGLPaaaaaaaaa@9CD1@ (5.3)

0 M x f( x;θ )dx=μ { θ 5 M+( M 4 +1 ) θ 4 +4 θ 3 M 3 +12 θ 2 M 2 +24θM+24 } e θM θ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaamyt aaqcLbsacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4bGaai 4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacaWGKbGaamiEaiabg2da 9iabeY7aTjabgkHiTKqbaoaalaaakeaajuaGdaGadaGcbaqcLbsacq aH4oqClmaaCaaajeaibeqaaKqzadGaaGynaaaajugibiaad2eacqGH RaWkjuaGdaqadaGcbaqcLbsacaWGnbqcfa4aaWbaaSqabKqaGeaaju gWaiaaisdaaaqcLbsacqGHRaWkcaaIXaaakiaawIcacaGLPaaajugi biabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey 4kaSIaaGinaiabeI7aXTWaaWbaaKqaGeqabaqcLbmacaaIZaaaaKqz GeGaamytaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey 4kaSIaaGymaiaaikdacqaH4oqClmaaCaaajeaibeqaaKqzadGaaGOm aaaajugibiaad2eajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaju gibiabgUcaRiaaikdacaaI0aGaeqiUdeNaamytaiabgUcaRiaaikda caaI0aaakiaawUhacaGL9baajugibiaadwgalmaaCaaajeaibeqaaK qzadGaeyOeI0IaeqiUdeNaamytaaaaaOqaaKqzGeGaeqiUdexcfa4a aeWabOqaceaagtrcLbsacqaH4oqClmaaCaaajeaibeqaaKqzadGaaG inaaaajugibiabgUcaRiaaiAdaaOGaayjkaiaawMcaaaaaaaa@940B@ (5.4)

Using equations (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 azlmaaBaaajeaibaqcLbmacaaIXaaajeaibeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaaaaa@3E2A@ 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 Pranav distribution are calculated as

δ 1 ( X )= 2{ θ 4 + θ 3 μ 3 +6 θ 2 μ 2 +18θμ+24 } e θμ θ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiaaikdajuaGdaGadaGcbaqcLbsacqaH4oqCjuaGdaahaaWc beqcbasaaKqzadGaaGinaaaajugibiabgUcaRiabeI7aXLqbaoaaCa aaleqajeaibaqcLbmacaaIZaaaaKqzGeGaeqiVd0wcfa4aaWbaaSqa bKqaGeaajugWaiaaiodaaaqcLbsacqGHRaWkcaaI2aGaeqiUdexcfa 4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqaH8oqBjuaGdaah aaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaigdacaaI4a GaeqiUdeNaeqiVd0Maey4kaSIaaGOmaiaaisdaaOGaay5Eaiaaw2ha aKqzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH4o qCcqaH8oqBaaaakeaajugibiabeI7aXLqbaoaabmaakeaajugibiab eI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaS IaaGOnaaGccaGLOaGaayzkaaaaaaaa@7D31@ (5.5)

δ 2 ( X )= 2{ θ 5 M+( M 4 +1 ) θ 4 +4 θ 3 M 3 +12 θ 2 M 2 +24θM+24 } e θM θ( θ 4 +6 ) μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqc LbsacaWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiaaikdajuaGdaGadaGcbaqcLbsacqaH4oqCjuaGdaahaaWc beqcbasaaKqzadGaaGynaaaajugibiaad2eacqGHRaWkjuaGdaqada GcbaqcLbsacaWGnbqcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqc LbsacqGHRaWkcaaIXaaakiaawIcacaGLPaaajugibiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGinaiab eI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaamytaK qbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGym aiaaikdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaju gibiaad2eajuaGdaahaaqcbawabKqaGeaajugWaiaaikdaaaqcLbsa cqGHRaWkcaaIYaGaaGinaiabeI7aXjaad2eacqGHRaWkcaaIYaGaaG inaaGccaGL7bGaayzFaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaa jugWaiabgkHiTiabeI7aXjaad2eaaaaakeaajugibiabeI7aXLqbao aabmaakeaajugibiabeI7aXTWaaWbaaKqaGeqabaqcLbmacaaI0aaa aKqzGeGaey4kaSIaaGOnaaGccaGLOaGaayzkaaaaaKqzGeGaeyOeI0 IaeqiVd0gaaa@8DFA@ (5.6)

Bonferroni and Lorenz curves

The Bonferroni and Lorenz curves Bonferroni8 and Gini indices have important used in economics to study income and poverty of any state. It’s relevance also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are obtained 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 qBaaqcfa4aa8qCaOqaaKqzGeGaamiEaiaadAgajuaGdaqadaGcbaqc LbsacaWG4baakiaawIcacaGLPaaaaKqaGeaajugWaiaaicdaaKqaGe aajugWaiaadghaaKqzGeGaey4kIipacaWGKbGaamiEaiabg2da9Kqb aoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamiCaiabeY7aTbaaju aGdaWadaGcbaqcfa4aa8qCaOqaaKqzGeGaamiEaiaadAgajuaGdaqa daGcbaqcLbsacaWG4baakiaawIcacaGLPaaajugibiaadsgacaWG4b GaeyOeI0cajeaibaqcLbmacaaIWaaajeaibaqcLbmacqGHEisPaKqz GeGaey4kIipajuaGdaWdXbGcbaqcLbsacaWG4bGaamOzaKqbaoaabm aakeaajugibiaadIhaaOGaayjkaiaawMcaaaqcbasaaKqzadGaamyC aaqcbasaaKqzadGaeyOhIukajugibiabgUIiYdGaamizaiaadIhaaO Gaay5waiaaw2faaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGym aaGcbaqcLbsacaWGWbGaeqiVd0gaaKqbaoaadmaakeaajugibiabeY 7aTjabgkHiTKqbaoaapehakeaajugibiaadIhacaWGMbqcfa4aaeWa aOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaajeaibaqcLbmacaWGXb aajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipacaWGKbGaamiEaaGc caGLBbGaayzxaaaaaa@9A3C@ (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 aGdaWdXbGcbaqcLbsacaWG4bGaamOzaKqbaoaabmaakeaajugibiaa dIhaaOGaayjkaiaawMcaaaqcbasaaKqzadGaaGimaaqcbasaaKqzad GaamyCaaqcLbsacqGHRiI8aiaadsgacaWG4bGaeyypa0tcfa4aaSaa aOqaaKqzGeGaaGymaaGcbaqcLbsacqaH8oqBaaqcfa4aamWaaOqaaK qbaoaapehakeaajugibiaadIhacaWGMbqcfa4aaeWaaOqaaKqzGeGa amiEaaGccaGLOaGaayzkaaqcLbsacaWGKbGaamiEaiabgkHiTaqcba saaKqzadGaaGimaaqcbasaaKqzadGaeyOhIukajugibiabgUIiYdqc fa4aa8qCaOqaaKqzGeGaamiEaiaadAgajuaGdaqadaGcbaqcLbsaca WG4baakiaawIcacaGLPaaaaKqaGeaajugWaiaadghaaKqaGeaajugW aiabg6HiLcqcLbsacqGHRiI8aiaadsgacaWG4baakiaawUfacaGLDb aajugibiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa eqiVd0gaaKqbaoaadmaakeaajugibiabeY7aTjabgkHiTKqbaoaape hakeaajugibiaadIhacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGc caGLOaGaayzkaaaajeaibaqcLbmacaWGXbaajeaibaqcLbmacqGHEi sPaKqzGeGaey4kIipacaWGKbGaamiEaaGccaGLBbGaayzxaaaaaa@9767@ (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 jkaiaawMcaaaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaamiCaaqc LbsacqGHRiI8aiaadsgacaWG4baaaa@544F@ (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 zkaaaajeaibaqcLbmacaaIWaaajeaibaqcLbmacaWGWbaajugibiab gUIiYdGaamizaiaadIhaaaa@5364@ (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 daqadaGcbaqcLbsacaWGWbaakiaawIcacaGLPaaaaKqaGeaajugWai aaicdaaKqaGeaajugWaiaaigdaaKqzGeGaey4kIipacaWGKbGaamiC aaaa@487A@ (6.5) and G=12 0 1 L( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaaGymaiabgkHiTiaaikdajuaGdaWdXbGcbaqcLbsacaWG mbqcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaaajeaiba qcLbmacaaIWaaajeaibaqcLbmacaaIXaaajugibiabgUIiYdGaamiz aiaadchaaaa@4945@ (6.6) respectively.

 Using pdf of Pranav distribution (1.1), it can be written

q xf( x;θ ) dx= { θ 5 q+ θ 4 ( q 4 +1)+4 θ 3 q 3 +12 θ 2 q 2 +24θq+24 } e θμ θ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaiaadAgajuaGdaqadaGcbaqcLbsacaWG4bGaai4o aiabeI7aXbGccaGLOaGaayzkaaaajeaibaqcLbmacaWGXbaajqwaa+ FaaKqzGcGaeyOhIukajugibiabgUIiYdGaamizaiaadIhacqGH9aqp juaGdaWcaaGcbaqcfa4aaiWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaS qabKqaGeaajugWaiaaiwdaaaqcLbsacaWGXbGaey4kaSIaeqiUdexc fa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacaGGOaGaamyCaK qbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGym aiaacMcacqGHRaWkcaaI0aGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaiodaaaqcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaa iodaaaqcLbsacqGHRaWkcaaIXaGaaGOmaiabeI7aXLqbaoaaCaaale qajeaibaqcLbmacaaIYaaaaKqzGeGaamyCaKqbaoaaCaaaleqajeai baqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaiaaisdacqaH4oqCca WGXbGaey4kaSIaaGOmaiaaisdaaOGaay5Eaiaaw2haaKqzGeGaamyz aKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH4oqCcqaH8oqBaa aakeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeI7aXTWaaWba aeqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaaGccaGLOa Gaayzkaaaaaaaa@9459@ (6.7)

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

B( p )= 1 p [ 1 { θ 5 q+ θ 4 ( q 4 +1)+4 θ 3 q 3 +12 θ 2 q 2 +24θq+24 } e θμ ( θ 4 +6 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadchaaaqcfa 4aamWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajuaGdaGa daGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaa aajugibiaadghacqGHRaWkcqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaGinaaaajugibiaacIcacaWGXbqcfa4aaWbaaSqabKqaGeaaju gWaiaaisdaaaqcLbsacqGHRaWkcaaIXaGaaiykaiabgUcaRiaaisda cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiaadg hajuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabgUcaRiaa igdacaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaa qcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsa cqGHRaWkcaaIYaGaaGinaiabeI7aXjaadghacqGHRaWkcaaIYaGaaG inaaGccaGL7bGaayzFaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaa jugWaiabgkHiTiabeI7aXjabeY7aTbaaaOqaaKqbaoaabmaakeaaju gibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGa ey4kaSIaaGOnaaGccaGLOaGaayzkaaaaaaGaay5waiaaw2faaaaa@8BBC@ (6.8)

and L( p )=1 { θ 5 q+ θ 4 ( q 4 +1)+4 θ 3 q 3 +12 θ 2 q 2 +24θq+24 } e θμ ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqbaoaacmaakeaajugibi abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaamyC aiabgUcaRiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaK qzGeGaaiikaiaadghajuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaa jugibiabgUcaRiaaigdacaGGPaGaey4kaSIaaGinaiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaamyCaKqbaoaaCaaa leqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGymaiaaikdacq aH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiaadgha juaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaik dacaaI0aGaeqiUdeNaamyCaiabgUcaRiaaikdacaaI0aaakiaawUha caGL9baajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0 IaeqiUdeNaeqiVd0gaaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdexc fa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2a aakiaawIcacaGLPaaaaaaaaa@852D@ (6.9)

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

B=1 { θ 5 q+ θ 4 ( q 4 +1)+4 θ 3 q 3 +12 θ 2 q 2 +24θq+24 } e θμ ( θ 4 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb Gaeyypa0JaaGymaiabgkHiTKqbaoaalaaakeaajuaGdaGadaGcbaqc LbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibi aadghacqGHRaWkcqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGin aaaajugibiaacIcacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaais daaaqcLbsacqGHRaWkcaaIXaGaaiykaiabgUcaRiaaisdacqaH4oqC juaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiaadghajuaGda ahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabgUcaRiaaigdacaaI YaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsaca WGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWk caaIYaGaaGinaiabeI7aXjaadghacqGHRaWkcaaIYaGaaGinaaGcca GL7bGaayzFaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugWaiab gkHiTiabeI7aXjabeY7aTbaaaOqaaKqbaoaabmaakeaajugibiabeI 7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIa aGOnaaGccaGLOaGaayzkaaaaaaaa@80E5@ (6.10)

G= 2{ θ 5 q+ θ 4 ( q 4 +1)+4 θ 3 q 3 +12 θ 2 q 2 +24θq+24 } e θμ ( θ 4 +6 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmaKqbaoaacmaakeaajugi biabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaam yCaiabgUcaRiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaa aKqzGeGaaiikaiaadghajuaGdaahaaWcbeqcbasaaKqzadGaaGinaa aajugibiabgUcaRiaaigdacaGGPaGaey4kaSIaaGinaiabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaamyCaKqbaoaaCa aaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGymaiaaikda cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiaadg hajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaa ikdacaaI0aGaeqiUdeNaamyCaiabgUcaRiaaikdacaaI0aaakiaawU hacaGL9baajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOe I0IaeqiUdeNaeqiVd0gaaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUde xcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI 2aaakiaawIcacaGLPaaaaaqcLbsacqGHsislcaaIXaaaaa@82C4@ (6.11)

Order statistics and renyi entropy measure

Order statistics

Let X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb qcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacaGGSaGaamiw aKqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaaiilaiaac6 cacaGGUaGaaiOlaiaacYcacaWGybqcfa4aaSbaaKqaGeaajugWaiaa d6gaaSqabaaaaa@4700@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ from Pranav distribution (1.1). Let X ( 1 ) < X ( 2 ) <...< X ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb qcfa4aaSbaaKqaGeaalmaabmaajeaibaqcLbmacaaIXaaajeaicaGL OaGaayzkaaaaleqaaKqzGeGaeyipaWJaamiwaSWaaSbaaKqaGeaalm aabmaajeaibaqcLbmacaaIYaaajeaicaGLOaGaayzkaaaabeaajugi biabgYda8iaac6cacaGGUaGaaiOlaiabgYda8iaadIfalmaaBaaaje aibaWcdaqadaqcbasaaKqzadGaamOBaaqcbaIaayjkaiaawMcaaaqa baaaaa@4C98@ denote the corresponding order statistics. The pdf and the cdf of the Kth order statistic, say Y= X ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb Gaeyypa0JaamiwaSWaaSbaaKqaGeaalmaabmaajeaibaqcLbmacaWG RbaajeaicaGLOaGaayzkaaaabeaaaaa@3DA2@ 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 gkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaaiyiaKqbaoaabmaake aajugibiaad6gacqGHsislcaWGRbaakiaawIcacaGLPaaajugibiaa cgcaaaGaamOraKqbaoaaCaaaleqajeaibaqcLbmacaWGRbGaeyOeI0 IaaGymaaaajuaGdaqadaGcbaqcLbsacaWG5baakiaawIcacaGLPaaa juaGdaGadaGcbaqcLbsacaaIXaGaeyOeI0IaamOraKqbaoaabmaake aajugibiaadMhaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaKqbaoaa CaaaleqajeaibaqcLbmacaWGUbGaeyOeI0Iaam4AaaaajugibiaadA gajuaGdaqadaGcbaqcLbsacaWG5baakiaawIcacaGLPaaaaaa@6C90@

= 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 yiaKqbaoaabmaakeaajugibiaad6gacqGHsislcaWGRbaakiaawIca caGLPaaajugibiaacgcaaaqcfa4aaabCaOqaaKqbaoaabmaakeaaju gibuaabeqaceaaaOqaaKqzGeGaamOBaiabgkHiTiaadUgaaOqaaKqz GeGaamiBaaaaaOGaayjkaiaawMcaaaqcbasaaKqzadGaamiBaiabg2 da9iaaicdaaKqaGeaajugWaiaad6gacqGHsislcaWGRbaajugibiab ggHiLdqcfa4aaeWaaOqaaKqzGeGaeyOeI0IaaGymaaGccaGLOaGaay zkaaqcfa4aaWbaaSqabKqaGeaajugWaiaadYgaaaqcLbsacaWGgbqc fa4aaWbaaSqabKqaGeaajugWaiaadUgacqGHRaWkcaWGSbGaeyOeI0 IaaGymaaaajuaGdaqadaGcbaqcLbsacaWG5baakiaawIcacaGLPaaa jugibiaadAgajuaGdaqadaGcbaqcLbsacaWG5baakiaawIcacaGLPa aaaaa@730C@

 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 qcfa4aaSbaaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamyEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaaeWbGcba qcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsacaWGUbaakeaa jugibiaadQgaaaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadQgacq GH9aqpcaWGRbaajeaibaqcLbmacaWGUbaajugibiabggHiLdGaamOr aKqbaoaaCaaaleqajeaibaqcLbmacaWGQbaaaKqbaoaabmaakeaaju gibiaadMhaaOGaayjkaiaawMcaaKqbaoaacmaakeaajugibiaaigda cqGHsislcaWGgbqcfa4aaeWaaOqaaKqzGeGaamyEaaGccaGLOaGaay zkaaaacaGL7bGaayzFaaqcfa4aaWbaaSqabKqaGeaajugWaiaad6ga cqGHsislcaWGQbaaaaaa@652B@

= 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 GaayzkaaaajeaibaqcLbmacaWGSbGaeyypa0JaaGimaaqcbasaaKqz adGaamOBaiabgkHiTiaadQgaaKqzGeGaeyyeIuoajuaGdaqadaGcba qcLbsafaqabeGabaaakeaajugibiaad6gacqGHsislcaWGQbaakeaa jugibiaadYgaaaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadQgacq GH9aqpcaWGRbaajeaibaqcLbmacaWGUbaajugibiabggHiLdqcfa4a aeWaaOqaaKqzGeGaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa4aaW baaSqabKqaGeaajugWaiaadYgaaaqcLbsacaWGgbqcfa4aaWbaaSqa bKqaGeaajugWaiaadQgacqGHRaWkcaWGSbaaaKqbaoaabmaakeaaju gibiaadMhaaOGaayjkaiaawMcaaaaa@6BC6@

 respectively, for k=1,2,3,...,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2 da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6cacaGG UaGaaiOlaiaacYcacaWGUbaaaa@3FE9@ .

 Thus, the pdf and the cdf of Kth order statistic of Pranav distribution (1.1) are obtained as

f Y ( y )= n! θ 4 ( θ+ x 3 ) e θx ( θ 4 +6 )( k1 )!( nk )! l=0 nk ( nk l ) ( 1 ) l             × [ 1 { θx( θ 2 x 2 +3θx+6 )+( θ 4 +6 ) } e θx ( θ 4 +6) ] k+l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi aadAgajuaGdaWgaaqcbasaaKqzadGaamywaaWcbeaajuaGdaqadaGc baqcLbsacaWG5baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaala aakeaajugibiaad6gacaGGHaGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaisdaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaam iEaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaaGccaGLOaGaayzk aaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabeI 7aXjaadIhaaaaakeaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaah aaWcbeqcbasaaKqzadGaaGinaaaajugibiabgUcaRiaaiAdaaOGaay jkaiaawMcaaKqbaoaabmaakeaajugibiaadUgacqGHsislcaaIXaaa kiaawIcacaGLPaaajugibiaacgcajuaGdaqadaGcbaqcLbsacaWGUb GaeyOeI0Iaam4AaaGccaGLOaGaayzkaaqcLbsacaGGHaaaaKqbaoaa qahakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiaad6 gacqGHsislcaWGRbaakeaajugibiaadYgaaaaakiaawIcacaGLPaaa aKqaGeaajugWaiaadYgacqGH9aqpcaaIWaaajeaibaqcLbmacaWGUb GaeyOeI0Iaam4AaaqcLbsacqGHris5aKqbaoaabmaakeaajugibiab gkHiTiaaigdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLb macaWGSbaaaaGcbaqcLbsaqaaaaaaaaaWdbiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOa8aacqGHxdaTjuaGdaWadaGcbaqcLbsacaaIXaGaeyOeI0sc fa4aaSaaaOqaaKqbaoaacmaakeaajugibiabeI7aXjaadIhajuaGda qadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOm aaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaju gibiabgUcaRiaaiodacqaH4oqCcaWG4bGaey4kaSIaaGOnaaGccaGL OaGaayzkaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibiabgUcaRiaaiAda aOGaayjkaiaawMcaaaGaay5Eaiaaw2haaKqzGeGaamyzaKqbaoaaCa aaleqajeaibaqcLbmacqGHsislcqaH4oqCcaWG4baaaaGcbaqcLbsa caGGOaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLb sacqGHRaWkcaaI2aGaaiykaaaaaOGaay5waiaaw2faaKqbaoaaCaaa leqajeaibaqcLbmacaWGRbGaey4kaSIaamiBaiabgkHiTiaaigdaaa aaaaa@D818@

and

F Y ( y )= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l [ 1 { θx( θ 2 x 2 +3θx+6 )+( θ 4 +6 ) } e θx ( θ 4 +6) ] j+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamyEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaaeWbGcba qcfa4aaabCaOqaaKqbaoaabmaakeaajugibuaabeqaceaaaOqaaKqz GeGaamOBaaGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaaajeaiba qcLbmacaWGSbGaeyypa0JaaGimaaqcbasaaKqzadGaamOBaiabgkHi TiaadQgaaKqzGeGaeyyeIuoajuaGdaqadaGcbaqcLbsafaqabeGaba aakeaajugibiaad6gacqGHsislcaWGQbaakeaajugibiaadYgaaaaa kiaawIcacaGLPaaaaKqaGeaajugWaiaadQgacqGH9aqpcaWGRbaaje aibaqcLbmacaWGUbaajugibiabggHiLdqcfa4aaeWaaOqaaKqzGeGa eyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaaju gWaiaadYgaaaqcfa4aamWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaa laaakeaajuaGdaGadaGcbaqcLbsacqaH4oqCcaWG4bqcfa4aaeWaaO qaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqc LbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacq GHRaWkcaaIZaGaeqiUdeNaamiEaiabgUcaRiaaiAdaaOGaayjkaiaa wMcaaKqzGeGaey4kaSscfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaW baaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaakiaa wIcacaGLPaaaaiaawUhacaGL9baajugibiaadwgajuaGdaahaaWcbe qcbasaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaOqaaKqzGeGaaiik aiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey 4kaSIaaGOnaiaacMcaaaaakiaawUfacaGLDbaajuaGdaahaaWcbeqc basaaKqzadGaamOAaiabgUcaRiaadYgaaaaaaa@A70A@

Renyi entropy measure

A popular entropy measure is given by Renyi entropy.9 If 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>0  and  γ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH+aGpcaaIWaGcqaaaaaaaaaWdbiaacckacaGGGcqcLbsapaGa aeyyaiaab6gacaqGKbGcpeGaaiiOaiaacckajugib8aacqaHZoWzcq GHGjsUcaaIXaaaaa@46E3@ .

Thus, the Renyi entropy for Pranav distribution (1.1) can be obtained as

T R ( γ )= 1 1γ log[ 0 θ 4γ ( θ 4 +6 ) γ ( θ+ x 3 ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaKqaGeaajugWaiaadkfaaSqabaqcfa4aaeWaaOqaaKqz GeGaeq4SdCgakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGymaiabgkHiTiabeo7aNbaaciGG SbGaai4BaiaacEgajuaGdaWadaGcbaqcfa4aa8qCaOqaaKqbaoaala aakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aGa eq4SdCgaaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaS qabeaajugibiaaisdaaaGaey4kaSIaaGOnaaGccaGLOaGaayzkaaqc fa4aaWbaaSqabKqaGeaajugWaiabeo7aNbaaaaqcfa4aaeWaaOqaaK qzGeGaeqiUdeNaey4kaSIaamiEaKqbaoaaCaaaleqajeaibaqcLbma caaIZaaaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWai abeo7aNbaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOe I0IaeqiUdeNaeq4SdCMaamiEaaaajugibiaadsgacaWG4baajeaiba qcLbmacaaIWaaajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipaaOGa ay5waiaaw2faaaaa@7F32@

= 1 1γ log[ 0 θ 4γ ( θ 4 +6 ) γ θ γ ( 1+ x 3 θ ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaape hakeaajuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGinaiabeo7aNbaaaOqaaKqbaoaabmaakeaajugibiabeI 7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIa aGOnaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiabeo 7aNbaaaaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaeq4S dCgaaKqbaoaabmaakeaajugibiaaigdacqGHRaWkjuaGdaWcaaGcba qcLbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaaakeaa jugibiabeI7aXbaaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaiba qcLbmacqaHZoWzaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugW aiabgkHiTiabeI7aXjabeo7aNjaadIhaaaqcLbsacaWGKbGaamiEaa qcbasaaKqzadGaaGimaaqcbasaaKqzadGaeyOhIukajugibiabgUIi YdaakiaawUfacaGLDbaaaaa@8062@

= 1 1γ log[ 0 θ 5γ ( θ 4 +6 ) γ j=0 ( γ j ) ( x 3 θ ) j e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaape hakeaajuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGynaiabeo7aNbaaaOqaaKqbaoaabmaakeaajugibiabeI 7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIa aGOnaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiabeo 7aNbaaaaqcfa4aaabCaOqaaKqbaoaabmaakeaajugibuaabeqaceaa aOqaaKqzGeGaeq4SdCgakeaajugibiaadQgaaaaakiaawIcacaGLPa aaaKqaGeaajugWaiaadQgacqGH9aqpcaaIWaaajeaibaqcLbmacqGH EisPaKqzGeGaeyyeIuoajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGe GaamiEaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaaGcbaqcLbsa cqaH4oqCaaaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzad GaamOAaaaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOe I0IaeqiUdeNaeq4SdCMaamiEaaaajugibiaadsgacaWG4baajeaiba qcLbmacaaIWaaajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipaaOGa ay5waiaaw2faaaaa@8830@

= 1 1γ log[ j=0 ( γ j ) θ 5γ ( θ 2 +6 ) γ θ j 0 e θγx x 3j dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaaqa hakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiabeo7a NbGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaaajeaibaqcLbmaca WGQbGaeyypa0JaaGimaaqcbasaaKqzadGaeyOhIukajugibiabggHi Ldqcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaiwdacqaHZoWzaaaakeaajuaGdaqadaGcbaqcLbsacqaH4oqC juaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaiA daaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaqbaqcLboacqaHZoWz aaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaamOAaaaaaa qcfa4aa8qCaOqaaKqzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbma cqGHsislcqaH4oqCcqaHZoWzcaWG4baaaKqzGeGaamiEaKqbaoaaCa aaleqajeaibaqcLbmacaaIZaGaamOAaaaajugibiaadsgacaWG4baa jeaibaqcLbmacaaIWaaajeaibaqcLbmacqGHEisPaKqzGeGaey4kIi paaOGaay5waiaaw2faaaaa@8680@

= 1 1γ log[ j=0 ( γ j ) θ 5γj ( θ 2 +6 ) γ 0 e θγx x 3j+11 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaaqa hakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiabeo7a NbGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaaajeaibaqcLbmaca WGQbGaeyypa0JaaGimaaqcbasaaKqzadGaeyOhIukajugibiabggHi Ldqcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaiwdacqaHZoWzcqGHsislcaWGQbaaaaGcbaqcfa4aaeWaaOqa aKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLb sacqGHRaWkcaaI2aaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasa aKqzadGaeq4SdCgaaaaajuaGdaWdXbGcbaqcLbsacaWGLbWcdaahaa qcbasabeaajugWaiabgkHiTiabeI7aXjabeo7aNjaadIhaaaqcLbsa caWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaiodacaWGQbGaey4kaS IaaGymaiabgkHiTiaaigdaaaqcLbsacaWGKbGaamiEaaqcbasaaKqz adGaaGimaaqcbasaaKqzadGaeyOhIukajugibiabgUIiYdaakiaawU facaGLDbaaaaa@858C@

= 1 1γ log[ j=0 ( γ j ) θ 5γ4j1 ( θ 4 +6 ) γ Γ( 3j+1 ) ( γ ) 3j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaigdacqGHsisl cqaHZoWzaaGaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqbaoaaqa hakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiabeo7a NbGcbaqcLbsacaWGQbaaaaGccaGLOaGaayzkaaqcfa4aaSaaaOqaaK qzGeGaeqiUde3cdaahaaqcbasabeaajugWaiaaiwdacqaHZoWzcqGH sislcaaI0aGaamOAaiabgkHiTiaaigdaaaaakeaajuaGdaqadaGcba qcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugi biabgUcaRiaaiAdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaiba qcLbmacqaHZoWzaaaaaaqcbasaaKqzadGaamOAaiabg2da9iaaicda aKqaGeaajugWaiabg6HiLcqcLbsacqGHris5aKqbaoaalaaakeaaju gibiabfo5ahLqbaoaabmaakeaajugibiaaiodacaWGQbGaey4kaSIa aGymaaGccaGLOaGaayzkaaaabaqcfa4aaeWaaOqaaKqzGeGaeq4SdC gakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaG4maiaa dQgacqGHRaWkcaaIXaaaaaaaaOGaay5waiaaw2faaaaa@7E28@

Stress–strength reliability

The stress– strength has wide applications in almost all areas of knowledge especially in engineering such as structures, 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 Pranav distribution with parameter θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qClmaaBaaajeaibaqcLbmacaaIXaaajeaibeaaaaa@3AA4@ and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qClmaaBaaajeaibaqcLbmacaaIYaaajeaibeaaaaa@3AA5@  respectively. Then the stress–strength reliability R 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 caWGybGaeyypa0JaamiEaaGccaGLOaGaayzkaaaajeaibaqcLbmaca aIWaaajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipacaWGMbqcfa4a aSbaaKqaGeaajugWaiaadIfaaSqabaqcfa4aaeWaaOqaaKqzGeGaam iEaaGccaGLOaGaayzkaaqcLbsacaWGKbGaamiEaaaa@5D5B@

= 0 f( x; θ 1 , α 1 ) F( x; θ 2 , α 2 )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWdXbGcbaqcLbsacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiE aiaacUdacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaaju gibiaacYcacqaHXoqyjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaa aOGaayjkaiaawMcaaaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaey OhIukajugibiabgUIiYdGaamOraKqbaoaabmaakeaajugibiaadIha caGG7aGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcLb sacaGGSaGaeqySdewcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaa kiaawIcacaGLPaaajugibiaadsgacaWG4baaaa@61E2@

=1 θ 1 [ 360 θ 1 θ 2 2 +1080 θ 2 3 +144 θ 2 ( θ 1 + θ 2 ) 2 +6( θ 1 θ 2 3 + θ 2 4 +6) ( θ 1 + θ 2 ) 3 +6 θ 1 θ 2 2 ( θ 1 + θ 2 ) 4 +6 θ 1 θ 2 ( θ 1 + θ 2 ) 5 + θ 1 ( θ 2 4 +6) ( θ 1 + θ 2 ) 6 ] ( θ 1 4 +6 )( θ 2 4 +6 ) ( θ 1 + θ 2 ) 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaSba aKqaGeaajugWaiaaigdaaSqabaqcfa4aamWaaKqzGeabaeqakeaaju gibiaaiodacaaI2aGaaGimaiabeI7aXLqbaoaaBaaajeaibaqcLbma caaIXaaaleqaaKqzGeGaeqiUde3cdaWgaaqcbasaaKqzadGaaGOmaa qcbasabaWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGHRaWk caaIXaGaaGimaiaaiIdacaaIWaGaeqiUde3cdaqhaaqcbasaaKqzad GaaGOmaaqcbasaaKqzadGaaG4maaaajugibiabgUcaRiaaigdacaaI 0aGaaGinaiabeI7aXLqbaoaaBaaajeaqbaqcLboacaaIYaaaleqaaK qzGeGaaiikaiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqa aKqzGeGaey4kaSIaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaS qabaqcLbsacaGGPaqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaaa keaajugibiabgUcaRiaaiAdacaGGOaGaeqiUdexcfa4aaSbaaKqaGe aajugWaiaaigdaaSqabaqcLbsacqaH4oqClmaaDaaajeaibaqcLbma caaIYaaajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaeqiUdexcfa 4aa0baaKqaGeaajugWaiaaikdaaKqaGeaajugWaiaaisdaaaqcLbsa cqGHRaWkcaaI2aGaaiykaiaacIcacqaH4oqCjuaGdaWgaaqcbasaaK qzadGaaGymaaWcbeaajugibiabgUcaRiabeI7aXLqbaoaaBaaajeai baqcLbmacaaIYaaaleqaaKqzGeGaaiykaKqbaoaaCaaaleqajeaiba qcLbmacaaIZaaaaKqzGeGaey4kaSIaaGOnaiabeI7aXLqbaoaaBaaa jeaibaqcLbmacaaIXaaaleqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGe aajugWaiaaikdaaSqabaqcfa4aaWbaaSqabKqaafaajug4aiaaikda aaqcLbsacaGGOaGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaigdaaS qabaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGOm aaWcbeaajugibiaacMcajuaGdaahaaWcbeqcbasaaKqzadGaaGinaa aaaOqaaKqzGeGaey4kaSIaaGOnaiabeI7aXLqbaoaaBaaajeaibaqc LbmacaaIXaaaleqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWai aaikdaaSqabaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaSbaaKqa GeaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaa qcbasaaKqzadGaaGOmaaWcbeaaaOGaayjkaiaawMcaaKqbaoaaCaaa leqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaeqiUdexcfa4aaS baaKqaGeaajugWaiaaigdaaSqabaqcLbsacaGGOaGaeqiUdexcfa4a a0baaKqaGeaajugWaiaaikdaaKqaGeaajugWaiaaisdaaaqcLbsacq GHRaWkcaaI2aGaaiykaKqbaoaabmaakeaajugibiabeI7aXLqbaoaa BaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaeqiUdexcfa 4aaSbaaKqaGeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaajuaG daahaaWcbeqcbasaaKqzadGaaGOnaaaaaaGccaGLBbGaayzxaaaaba qcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aa0baaKqaGeaajugWaiaa igdaaKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aaakiaawI cacaGLPaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaqhaaqcbasa aKqzadGaaGOmaaqcbasaaKqzadGaaGinaaaajugibiabgUcaRiaaiA daaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaa BaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaeqiUdexcfa 4aaSbaaKqaGeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaajuaG daahaaWcbeqcbasaaKqzadGaaG4naaaaaaaaaa@1D05@ .

Parameters estimation

Method of Moments Estimates (MOME) of parameters

Method of moments can be calculated equating population mean of Pranav distribution to the sample mean, which is as follows:

MOME of θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaacaaaa@384A@ θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ can be obtained as

θ 5 x ¯ θ 4 +6θ x ¯ 24=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiqadIhagaqe aiabgkHiTiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaK qzGeGaey4kaSIaaGOnaiabeI7aXjqadIhagaqeaiabgkHiTiaaikda caaI0aGaeyypa0JaaGimaaaa@4B48@ (9.1)

Maximum likelihood estimates (mle) of parameters

Let ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiEaKqbaoaaBaaajqwaa+FaaKqzGcGaaGymaaWcbeaa jugibiaacYcacaWG4bqcfa4aaSbaaKqaGeaajugWaiaaikdaaSqaba qcLbsacaGGSaGaamiEaKqbaoaaBaaajeaibaqcLbmacaaIZaaaleqa aKqzGeGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWG4bqcfa4aaS baaKqaafaajug4aiaad6gaaSqabaaakiaawIcacaGLPaaaaaa@5079@ be a random sample of size n from (1.1). The likelihood function, L of Pranav distribution is given by

L= ( θ 4 θ 4 +6 ) n i=1 n ( θ+ x i 3 ) e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0tcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaI0aaaaaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGinaaaajugibiabgUcaRiaaiAda aaaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaamOBaa aajuaGdaqeWbGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIa amiEaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqbaoaaCaaale qajeaibaqcLbmacaaIZaaaaaGccaGLOaGaayzkaaaajeaibaqcLbma caWGPbGaeyypa0JaaGymaaqcbasaaKqzadGaamOBaaqcLbsacqGHpi s1aiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaamOBaiab eI7aXjqadIhagaqeaaaaaaa@67F5@

and its log likelihood function is thus obtained as

lnL=nln( θ 4 θ 4 +6 )+ i=1 n ln( θ+ x i 3 ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaiaadYeacqGH9aqpcaWGUbGaciiBaiaac6gajuaGdaqadaGc baqcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaisdaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqc LbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaaaaaOGaayjkaiaawMcaaK qzGeGaey4kaSscfa4aaabCaOqaaKqzGeGaciiBaiaac6gajuaGdaqa daGcbaqcLbsacqaH4oqCcqGHRaWkcaWG4bqcfa4aaSbaaSqaaKqzGe GaamyAaaWcbeaajuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaaaOGa ayjkaiaawMcaaaqcbauaaKqzGdGaamyAaiabg2da9iaaigdaaKqaGe aajugWaiaad6gaaKqzGeGaeyyeIuoacqGHsislcaWGUbGaeqiUdeNa bmiEayaaraaaaa@6A24@ (9.2)

The maximum likelihood estimates (MLEs) 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@ ,

lnL θ = 4n θ 3nθ ( θ 4 +6) n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaciiBaiaac6gacaWGmbaakeaajugibiabgkGi 2kabeI7aXbaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaI0aGaamOBaa GcbaqcLbsacqaH4oqCaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaaG4m aiaad6gacqaH4oqCaOqaaKqzGeGaaiikaiabeI7aXTWaaWbaaKqaGe qabaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGOnaiaacMcaaaGaeyOe I0IaamOBaiqadIhagaqeaiabg2da9iaaicdaaaa@5795@ (9.3)

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b Gbaebaaaa@379A@ is the sample mean.

The equation (9.3) can be solved directly to estimate the value of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ ,

R–Software is used to compute the value of parameter.

A simulation study

A Simulation study has been performed in the present section, it consists in generating N=10,000 pseudo–random samples of sizes 10, 20, 30, 40, 50 from Pranav distribution. Acceptance and rejection method has been used for the simulation of data. Average bias and mean square error of the MLEs of the parameter  are estimated using the following formulae

 Average Bias = 1 N j=1 n ( θ j θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaamOtaaaajuaGdaaeWbGcbaqcLbsa caGGOaGafqiUdeNbambajuaGdaWgaaqcbasaaKqzadGaamOAaaWcbe aaaKqaGeaajugWaiaadQgacqGH9aqpcaaIXaaajeaibaqcLbmacaWG UbaajugibiabggHiLdGaeyOeI0IaeqiUdeNaaiykaaaa@4C49@ , MSE= 1 N j=1 n ( θ j θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaamOtaaaajuaGdaaeWbGcbaqcLbsa caGGOaGafqiUdeNbambajuaGdaWgaaqcbasaaKqzadGaamOAaaWcbe aaaKqaGeaajugWaiaadQgacqGH9aqpcaaIXaaajeaibaqcLbmacaWG UbaajugibiabggHiLdGaeyOeI0IaeqiUdeNaaiykaKqbaoaaCaaale qajeaibaqcLbmacaaIYaaaaaaa@4F18@

The following algorithm can be used to generate random sample from Pranav distribution. The process to generate a random sample consists of running the algorithm as often as necessary, say n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@ times.

Rejection Method for Continuous distribution.10

Suppose Rejection method is used in rectangular area (which can be plot using target density function) to cover the target density   f X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaaiiOaiaadAgajuaGpaWaaSbaaKqaGeaajugWa8qacaWG ybaal8aabeaaaaa@3D4F@ , then generates candidate points uniformly within the rectangle area. h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAai aacIcacaWG4bGaaiykaaaa@39BD@ is candidate density (which is known density function like uniform, exponential, etc.). However if the rectangular area is infinite, then it can cannot generate points uniformly within it, because it has infinite area. Instead it needs a shape with finite area, within which we can simulate points uniformly.

Let X have pdf h and, given X, let Y~U(0,kh(X)) then (X,Y) is uniformly distributed over the region A (say area) defined by the curve kh and 0 are highest and lowest points respectively.

It is noted that the range of Y depends on X.

It can be use conditional probability for example:

P((X,Y)(x,x+dx)*(y,y+dy)) =P(Y(y,y+dy)/X(x,x+dx))P(X(x,x+dx)) = dy kh(x) h(x)dx = 1 k dxdy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGqbGaaiikaiaacIcacaWGybGaaiilaiaacMfacaGGPaGaeyicI4Sa aiikaiaadIhacaGGSaGaamiEaiabgUcaRiaadsgacaWG4bGaaiykaK qzadGaaiOkaKqzGeGaaiikaiaadMhacaGGSaGaamyEaiabgUcaRiaa dsgacaWG5bGaaiykaiaacMcaaOqaaKqzGeGaeyypa0JaaiiuaiaacI cacaGGzbGaeyicI4SaaiikaiaadMhacaGGSaGaamyEaiabgUcaRiaa dsgacaWG5bGaaiykaiaac+cacaWGybGaeyicI4SaaiikaiaadIhaca GGSaGaamiEaiabgUcaRiaadsgacaWG4bGaaiykaiaacMcacaWGqbGa aiikaiaadIfacaGGOaGaamiEaiaacYcacaWG4bGaey4kaSIaamizai aadIhacaGGPaGaaiykaaGcbaqcLbsacqGH9aqpjuaGdaWcaaGcbaqc LbsacaWGKbGaamyEaaGcbaqcLbsacaWGRbGaamiAaiaacIcacaWG4b GaaiykaaaacaWGObGaaiikaiaadIhacaGGPaGaamizaiaadIhaaOqa aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsaca WGRbaaaiaadsgacaWG4bGaamizaiaadMhaaaaa@8645@

The chance of being in a small rectangle of size dx *dy is the same anywhere in A.

Algorithm

Rejection method: To simulate from the density fx, it is assumed that envelope density h from which it can simulate, and that have some k< MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgacqGH8a apcqGHEisPaaa@3950@ such that sup x f X (x) h(x) k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaci4Cai aacwhacaGGWbWcdaWgaaqcbasaaKqzadGaamiEaaqcbasabaqcfa4a aSaaaOqaaKqzGeGaamOzaKqbaoaaBaaajeaibaqcLbmacaWGybaale qaaKqzGeGaaiikaiaadIhacaGGPaaakeaajugibiaadIgacaGGOaGa amiEaiaacMcaaaGaeyizImQaam4Aaaaa@4A82@ Simulate X from h. where h(x)=θ e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAai aacIcacaWG4bGaaiykaiabg2da9iabeI7aXjaacwgalmaaCaaajeai beqaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@4287@

  1. Generate Y~U(0,kh(X)) , where k= θ 4 ( θ 4 +6) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4Aai abg2da9KqbaoaalaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaI0aaaaaGcbaqcLbsacaGGOaGaeqiUdexcfa4aaWbaaS qabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI2aGaaiykaaaa aaa@46D8@
  2. If Y< f X (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamywai abgYda8iaadAgajuaGdaWgaaqcKfaG=haajugOaiaadIfaaKqaGgqa aKqzGeGaaiikaiaadIhacaGGPaaaaa@415D@ then return X, otherwise go back to step 1.

The average bias (mean square error) of simulated estimate of parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ for different values of n and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ are presented in Table 1.

n

Parameter θ

0.05

0.05

1

2

10

0.02099(0.004406)

0.110315(0.121694)

0.114163(0.130332)

0.107694(0.11598)

20

0.010726(0.002301)

0.05524(0.061028)

0.058333(0.068055)

0.05615(0.063075)

30

0.006501(0.001268)

0.034787(0.036304)

0.036529 (0.04003)

0.03291(0.032500)

40

0.004479(0.0008025)

0.024951(0.024903)

0.025633(0.026284)

0.021125(0.01785)

50

0.003482(0.0006063)

0.016715(0.019348)

0.020210(0.020423)

0.016287(0.01326)

Table 1 Average bias (mean square error) of the simulated estimates of parameter θ

The graphs of estimated mean square error of the MLE for different values of parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@ have been presented in Figure 6.

Figure 6 Estimated mean squared error of the MLEs for different values of θ and n.

Illustrative example

Data set 1: Present data have been taken from Jones et al.,11 and it is associated with behavioral science. The detailed about data are given in Jones et al.11 A study conducted by the authors in a city located at the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis, which are:

19(16)    20(15)   21(14) 22(9)        23(12)   24(10)   25(6) 26(9)           27(8)     28(5)     29(6)      30(4)      31(3)     32(4) 33 34          35(4)      36(2)     37(2) 39                42 44

Data set 2: This data set is the strength data of glass of the aircraft window reported by:12
18.83      20.8     21.657   23.03      23.23      24.05      24.321   25.5      25.52      25.8        26.69     26.77   26.78    27.05      27.67      29.9      31.11     33.2        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

Data Set 3: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm taken from.13

 1.312     1.314      1.479      1.552      1.700      1.803      1.861      1.865      1.944      1.958      1.966      1.997      2.006      2.021      2.027      2.055      2.063      2.098      2.140      2.179      2.224      2.240      2.253      2.270      2.272      2.274      2.301      2.301      2.359      2.382      2.382      2.426      2.434      2.435      2.478      2.490      2.511      2.514      2.535      2.554      2.566      2.570      2.586      2.629      2.633      2.642      2.648      2.684      2.697      2.726      2.770      2.773      2.800      2.809      2.818      2.821      2.848      2.880      2.954      3.012      3.067      3.084      3.090      3.096      3.128      3.233      3.433      3.585      3.858     

From Above data sets, Pranav distribution has been fitted along with one parameter Sujatha, Ishita and and Akash, Shanker ,Lindley and exponential distributions. pdf, and cdf of these distributions are given in Table 2. The ML estimates, values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ and K–S statistics of the fitted distributions are shown in Table 3.

Distribution

pdf

cdf

Akash

f( t )= θ 3 θ 2 +2 ( 1+ t 2 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaG4maaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaikdaaaqcLbsacqGHRaWkcaaIYaaaaKqbaoaabmaakeaaju gibiaaigdacqGHRaWkcaWG0bqcfa4aaWbaaSqabKqaGeaajugWaiaa ikdaaaaakiaawIcacaGLPaaajugibiaadwgajuaGdaahaaWcbeqcba saaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@590B@

F( t )=1[ 1+ θt( θt+2 ) θ 2 +2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRK qbaoaalaaakeaajugibiabeI7aXjaadshajuaGdaqadaGcbaqcLbsa cqaH4oqCcaWG0bGaey4kaSIaaGOmaaGccaGLOaGaayzkaaaabaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiab gUcaRiaaikdaaaaakiaawUfacaGLDbaajugibiaadwgajuaGdaahaa WcbeqcbasaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@5C5A@

Shanker

f( t )= θ 2 θ 2 +1 ( θ+t ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaikdaaaqcLbsacqGHRaWkcaaIXaaaaKqbaoaabmaakeaaju gibiabeI7aXjabgUcaRiaadshaaOGaayjkaiaawMcaaKqzGeGaamyz aKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH4oqCcaWG4baaaa aa@5735@

F( t )=1[ 1+ θt θ 2 +1 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRK qbaoaalaaakeaajugibiabeI7aXjaadshaaOqaaKqzGeGaeqiUdexc fa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIXa aaaaGccaGLBbGaayzxaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaa jugWaiabgkHiTiabeI7aXjaadIhaaaaaaa@555C@

Sujatha

f( x;θ )= θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaW baaSqabKqaGeaajugWaiaaiodaaaaakeaajugibiabeI7aXTWaaWba aKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaS IaaGOmaaaajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIaamiEaiab gUcaRiaadIhalmaaCaaajeaibeqaaKqzadGaaGOmaaaaaOGaayjkai aawMcaaKqzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsisl cqaH4oqCcaWG4baaaKqzGeGaai4oaiaadIhacqGH+aGpcaaIWaGaai ilaiabeI7aXjabg6da+iaaicdaaaa@6718@

F( x,θ )=1[ 1+ θx( θx+θ+2 ) θ 2 +θ+2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacYcacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaaGymaiabgkHiTKqbaoaadmaakeaajugibi aaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacqaH4oqCcaWG4bqcfa4a aeWaaOqaaKqzGeGaeqiUdeNaamiEaiabgUcaRiabeI7aXjabgUcaRi aaikdaaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqa bKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkca aIYaaaaaGccaGLBbGaayzxaaqcLbsacaWGLbqcfa4aaWbaaSqabKqa GeaajugWaiabgkHiTiabeI7aXjaadIhaaaaaaa@63FC@

Ishita

f 0 ( x;θ )= θ 3 θ 3 +2 ( θ+ x 2 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaaicdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaawMcaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaiodaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLb macaaIZaaaaKqzGeGaey4kaSIaaGOmaaaajuaGdaqadaGcbaqcLbsa cqaH4oqCcqGHRaWkcaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaaakiaawIcacaGLPaaajugibiaadwgajuaGdaahaaWcbeqcbasa aKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@5F50@

F( x )=1[ 1+ θx( θx+2 ) θ 3 +2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRK qbaoaalaaakeaajugibiabeI7aXjaadIhajuaGdaqadaGcbaqcLbsa cqaH4oqCcaWG4bGaey4kaSIaaGOmaaGccaGLOaGaayzkaaaabaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiab gUcaRiaaikdaaaaakiaawUfacaGLDbaajugibiaadwgajuaGdaahaa WcbeqcbasaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@5C67@

Lindley

f( x;θ )= θ 2 θ+1 ( 1+x ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaW baaSqabKqaGeaajugWaiaaikdaaaaakeaajugibiabeI7aXjabgUca Riaaigdaaaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiaadIhaaO GaayjkaiaawMcaaKqzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbma cqGHsislcqaH4oqCcaWG4baaaaaa@5559@

F( x;θ )=1[ θ+1+θx θ+1 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaaGymaiabgkHiTKqbaoaadmaakeaajuaGda WcaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaGaey4kaSIaeqiUdeNa amiEaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaaaaGccaGLBbGaay zxaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiab eI7aXjaadIhaaaaaaa@5684@

Exponential

f( x;θ )=θ e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaeqiUdeNaamyzaKqbaoaaCaaaleqajeaiba qcLbmacqGHsislcqaH4oqCcaWG4baaaaaa@4784@

F( x;θ )=1 e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaaGymaiabgkHiTiaadwgajuaGdaahaaWcbe qcbasaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaa@4756@

Table 2 The pdf and the cdf of fitted distributions

Data Set

Model

Parameter estimate

-2ln L

AIC

BIC

K-S statistic

Data 1

Pranav

0.160222

945.03

947.03

948.94

0.362

Ishita

0.120083

980.02

982.02

984.62

0.399

Sujatha

0.117456

985.69

987.69

990.29

0.403

Akash

0.11961

981.28

983.28

986.18

0.4

Shanker

0.079746

1033.1

1035.1

1037.99

0.442

Lindley

0.077247

1041.64

1043.64

1046.54

0.448

Exponential

0.04006

1130.26

1132.26

1135.16

0.525

Data 2

Pranav

0.129818

232.77

234.77

236.68

0.253

Ishita

0.097325

240.48

242.48

244.39

0.298

Sujatha

0.09561

241.5

243.5

245.41

0.302

Akash

0.097062

240.68

242.68

244.11

0.266

Shanker

0.064712

252.35

254.35

255.78

0.326

Lindley

0.062988

253.99

255.99

257.42

0.333

Exponential

0.032455

274.53

276.53

277.96

0.426

Data 3

Pranav

1.225138

217.12

219.12

221.03

0.303

Ishita

0.931571

223.14

225.14

227.05

0.33

Sujatha

0.936119

221.6

223.6

225.52

0.364

Akash

0.964726

224.28

226.28

228.51

0.348

Shanker

0.658029

233.01

235.01

237.24

0.355

Lindley

0.659

238.38

240.38

242.61

0.39

Exponential

0.407941

261.74

263.74

265.97

0.434

Table 3 MLE’s, -2ln L, AIC, BIC, K-S Statistics of the fitted distributions of data-sets 1-3

Profile plots of MLE of parameter and fitted probability plots of proposed distributions, on considered data sets are given in the Figure 7–9 respectively.

Figure 7 Profile plot of parameter and fitted probability plot of distributions for dataset-1.
Figure 8 Profile plot of parameter and fitted probability plot of distributions for dataset-2.
Figure 9 Profile plot of parameter and fitted probability plot of distributions for dataset-3.

Concluding remarks

In this study, a new one parameter Pranav distribution has been proposed. Its mathematical properties including moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, and stress–strength reliability have been discussed. Simulation study of Pranav distribution has also been discussed. The method of moments and the method of maximum likelihood estimation have been derived for estimating the parameter. Finally, three numerical examples of real lifetime datasets form Biology have been presented to test the goodness of fit of the Pranav distribution. Its fit was found satisfactory over exponential, Lindley, Sujatha, Ishita, Akash and Shanker distributions. It can be considered as good model for biological study.

Note: The paper is named Pranav distribution in the name of my eldest son Pranav Shukla.

Acknowledgements

None.

Conflict of interest

Author declares that there is no conflict of interest.

References

  1. Lindley DV. Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Series B. 1958;20(1):102–107.
  2. Shanker R. Akash distribution and Its Applications. International Journal of Probability and Statistics. 2015;4(3):65–75.
  3. Shanker R. Shanker distribution and Its Applications. International Journal of Statistics and Applications. 2015;5(6):338–348.
  4. Shanker R, Shukla KK. Ishita distribution and Its Applications. BBIJ. 2017;5(2):1–9.
  5. Shanker R. Sujatha Distribution and Its Applications. Statistics in Transition–New series. 2016;17(3):1–20.
  6. Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its Applications. Mathematics Computing and Simulation. 2008;78:493–506.
  7. Shaked M, Shanthikumar JG. Stochastic Orders and Their Applications. New York: Academic Press; 1994.
  8. Bonferroni CE. Elementi di Statistca generale, Seeber, Firenze; 1930.
  9. Renyi entropy. On measures of entropy and information. Proc Fourth Berkeley Symp on Math Statist and Prob. 1961;(1):547–561.
  10. Balakrishnan N, Victor L, Antonio S. A mixture model based on Birnhaum–Saunders Distributions. A study conducted by Authors regarding the Scores of the GRASP (General Rating of Affective Symptoms for Preschoolers), in a city located at South Part of the Chile; 2010.
  11. Jones O, Maillardet R, Robinson A. Introduction to scientific Programming and Simulation using R. New York: Taylor & Francis Group; 2009.
  12. Fuller EJ, Frieman S, Quinn J. Fracture mechanics approach to the design of glass aircraft windows: A case study. SPIE Proc. 1994;(2286):419–430.
  13. Bader MG, Priest AM. Statistical aspects of fiber and bundle strength in hybrid composites, In; Hayashi T, Kawata K, Umekawa S, editors. Progress in Science in Engineering Composites, ICCM–IV, Tokyo. 1982:1129–1136.
©2018 Shukla. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.
© 2019 MedCrave Publishing, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use.
Creative Commons License Open Access by MedCrave Publishing is licensed under a Creative Commons Attribution 4.0 International License.
Based on a work at https://medcraveonline.com
Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version | Opera |Privacy Policy