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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Review Article Volume 13 Issue 3

An extended sujatha distribution with statistical properties and applications

Hosenur Rahman Prodhani, Rama Shanker

Department of Statistics, Assam University, Silchar, India

Correspondence: Rama Shanker, Department of Statistics, Assam University, Silchar, India

Received: September 21, 2024 | Published: September 12, 2024

Citation: Prodhani HR, Shanker R. An extended sujatha distribution with statistical properties and applications. Biom Biostat Int J. 2024;13(4):96-105. DOI: 10.15406/bbij.2024.13.00420

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Abstract

In this paper, an extended Sujatha distribution has been proposed using the exponentiated technique on Sujatha distribution. Statistical properties including survival function, hazard function, harmonic mean, moment generating function, order statistics and Renyi entropy have been discussed. Moments of the proposed distribution has been obtained. The estimation of parameters using the maximum likelihood method and maximum product spacing method has been explained. The simulation study has been presented to know the performance of maximum likelihood estimates as the sample size increases. Finally, two examples of real lifetime datasets from the engineering field have been presented to demonstrate its applications and the goodness of fit of extended Sujatha distribution shows better fit over exponentiated exponential and exponentiated Aradhana distributions.

Keywords: exponentiated distribution, sujatha distribution, statistical properties, maximum likelihood estimation, applications

Introduction

The exponentiation technique is a new concept of generalizing a given distribution which results into introducing an additional parameter in a distribution. Gupta et al.,1 proposed exponentiated technique of generalizing a new distribution and proposed exponentiated exponential distribution (EED) using exponentiated technique. This family came with scale and shape parameter, similar to Weibull or Gamma families. Later, some important statistical properties of EED studied by Gupta and Kundu.2 It has been observed that many characteristics of the new family were identical to those of the Weibull or Gamma families. Therefore, this distribution can be used as an alternative to Gamma or Weibull distributions. Two-parameter gamma and Weibull distributions are the most commonly used distributions for analyzing lifetime data. Gamma distributions have many applications beyond lifetime distributions. However, its major drawback is that its survival function cannot be obtained in a closed form unless the shape parameter is an integer. This makes the Gamma distribution a little less popular than the Weibull distribution, whose survival function and failure rate have very simple and easy-to-study forms. Presently exponentiated distributions and their mathematical properties are extensively studied for applied science experimental datasets. Pal et al.,3 studied the exponentiated Weibull family as an extension of Weibull distribution. Following the approach of deriving EED, during recent decades several researchers attempted to derive exponentiated version of many distributions. A Generalized Lindley distribution studied by Nadarajah et al.,4 the exponentiated generalized Lindley distribution (EGLD) studied by Rodrigues et al.,5 exponentiated Shanker distribution (ESHD) studied by Jayakumar and Elangovan,6 exponentiated Ishita distribution (EID) studied by Rather and Subramanian,7 exponentiated Aradhana distribution (EAD) studied by Ganaie and Rajagopalan.8

Sujatha distribution has been proposed by Shanker9 for analyzing lifetime data from engineering and biomedical science. The probability density function (pdf) and the cumulative distribution function (cdf) of Sujatha distribution can be expressed as

f(x)=θ3θ2+θ+2(1+x+x2)eθx;x>0,θ>0        (1)

F(x)=1 [1+θx(θx+θ+2)θ2+θ+2]eθx;   x>0,θ>0     (2)                                        

where θ is a scale parameter.

Statistical properties, estimation of parameter and applications of Sujatha distribution are studied by Shanker.9 Later on, several generalizations and modifications of Sujatha distribution have been done by several researchers. For example, quasi Sujatha distribution by Shanker,10 a generalization of Sujatha distribution by Shanker et al.,11 a two-parameter Sujatha distribution by Mussie and Shanker,12 a new two parameter Sujatha distribution by Mussie and Shanker,13 another new two parameter Sujatha distribution by Shanker,14 weighted Sujatha distribution by Shanker and Shukla,15 power Sujatha distribution by Shanker and Shukla,16 a new quasi Sujatha distribution by Shanker and Shukla,17 generalized Inverse power Sujatha distribution by Okoli et al.,18 and Marshal-Olkin Sujatha distribution by Ikechukwu and Eghwerideo19 are some among others.

The rationale behind the introduction of the extended Sujatha distribution presented in this paper can be elucidated as follows:

Consider a scenario where we have a series system comprised of independent components, and let X1,X2,...,Xα are independent random variables comes from a distribution with cdf F(x) and represent the failure times of the respective components. It is assumed that these components are independent of each other and identical. In this context, the probability that the entire system will experience failure before a specific time x can be expressed as follows:

Pr[max(X1,X2,...,Xα)x]=Pr(X1x)Pr(X2x)...Pr(Xαx)

=F(x)F(x)...F(x)

=[F(x)]α

Therefore, it provides the distribution that characterizes the failure of a series system consisting of independent components.

  1. Its pdf has unimodal and positively skewed shape, its hazard function has monotonically increased, monotonically decreased and L- shape.
  2. The proposed distribution has closed forms for cdf, survival function and hazard function.
  3. It can be considered a good distribution to fit positively skewed data that other popular lifetime distributions might not be able to fit appropriately.

Various statistical properties including hazard function, order statistics and Renyi entropy have been discussed. Estimation of parameters using maximum likelihood method and maximum product spacing method have discussed. The applications and the goodness of fit of the proposed distribution have been illustrated with two examples of observed real datasets from engineering. The proposed distribution gives much closure fit then exponentiated exponential and exponentiated Aradhana distributions.

An extended sujatha distribution

Following exponentiated method is use to propose the new distribution named as extended Sujatha distribution.

If f(x) and F(x) are pdf and cdf respectively of a random variable X, then the new proposed exponentiated family of distribution function has the following cdf and pdf

G(x)=[F(x)]α;x>0,α>0  (3)

g(x)=α[F(x)]α1f(x);x>0,α>0   (4)

Let X is a random variable which follows extended Sujatha distribution (ESD). The cdf of ESD can be obtained as

G(x;θ,α)=[1{1+θx(θx+θ+2)θ2+θ+2}eθx]α;x>0,θ>0,α>0      (5)

Thus, the pdf of ESD can be expressed as

g(x;θ,α)=αθ3θ2+θ+2(1+x+x2)eθx[1{1+θx(θx+θ+2)θ2+θ+2}eθx]α1;x>0,θ>0,α>0  (6)       

For α=1 the ESD reduced to Sujatha distribution. The nature of the pdf of ESD has been shown in the following figure 1.

Figure 1 Graphs of the pdf of ESD for selected values of the parameters.

It is obvious that for fixed value of the parameter α and for increasing values of parameter θ, the shape of ESD approaches approximately towards leptokurtic and positively skewed. On the other hands for fixed value of the parameter θ and for increasing values of parameter α, the shape of ESD approaches approximately towards positively skewed. The behaviors of the cdf of ESD have been shown in the figure 2.

Figure 2 Graphs of the cdf of ESD for selected values of the parameters.

Statistical properties of ESD

In this section, some important statistical properties of ESD have been studied.

Survival function

The Survival function of ESD can be obtained as

S(x)=1[1{1+θx(θx+θ+2)θ2+θ+2}eθx]α;x>0,θ>0,α>0        (7)

Hazard function

The hazard function of ESD can be obtained as

h(x)=αθ3θ2+θ+2(1+x+x2)eθx[1[1+θx(θx+θ+2)θ2+θ+2]eθx]α11[1[1+θx(θx+θ+2)θ2+θ+2]eθx]α;x>0,θ>0,α>0                   (8)

The natures of the hazard function of ESD are shown for varying values of parameters in the figure 3.

Figure 3 Graphs of hazard function of ESD for selected values of the parameters.

From the figure 3 we observed that when α is fixed and for all values of θ the hazard function is monotonically increasing and when θ is fixed and α < 1 then the hazard function is decreasing and after a certain time it is increasing and when θ is fixed and α < 1 then hazard function is a increasing.

The linear representation

Using the binomial expansion of (1x)n=i=0(1)i(ni)xi , we can be obtained the pdf of ESD as

g(x)=αi=0j=0k=0(1)i(α1i)(ij)(jk)θk+j+3(θ+2)jk(θ2+θ+2)j+1xj+k(1+x+x2)eθ(i+1)x                (9)

Moments of ESD

Using the linear representation in 9, the r th moments about origin μr' of ESD can be obtained as

μ r ' =E( X r )= 0 x r g(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaqhaaqaaKqzadGaamOCaaWcbaqcLbmacaGGNaaaaKqzGeGaeyyp a0JaamyraKqbaoaabmaakeaajugibiaadIfajuaGdaahaaWcbeqaaK qzGeGaamOCaaaaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aa8qC aOqaaKqzGeGaamiEaKqbaoaaCaaaleqabaqcLbsacaWGYbaaaiaadE gacaGGOaGaamiEaiaacMcacaWGKbGaamiEaaWcbaqcLbsacaaIWaaa leaajugibiabg6HiLcGaey4kIipaaaa@556F@

=α i=0 j=0 k=0 ( 1 ) i ( α1 i ) ( i j )( j k ) θ j+k+3 ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 0 x j+k+r ( 1+x+ x 2 ) e θ( i+1 )x dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpcqaHXoqyjuaGdaaeWbGcbaqcfa4aaabCaOqaaKqbaoaaqaha keaajuaGdaqadaGcbaqcLbsacqGHsislcaaIXaaakiaawIcacaGLPa aajuaGdaahaaWcbeqaaKqzGeGaamyAaaaaaSqaaKqzGeGaam4Aaiab g2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5aaWcbaqcLbsaca WGQbGaeyypa0JaaGimaaWcbaqcLbsacqGHEisPaiabggHiLdqcfa4a aeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsaqaaaaaaaaaWdbiabeg 7aHjabgkHiTiaaigdaaOWdaeaajugib8qacaWGPbaaaaGcpaGaayjk aiaawMcaaaWcbaqcLbsacaWGPbGaeyypa0JaaGimaaWcbaqcLbsacq GHEisPaiabggHiLdqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqc LbsapeGaamyAaaGcpaqaaKqzGeWdbiaadQgaaaaak8aacaGLOaGaay zkaaqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsapeGaamOA aaGcpaqaaKqzGeWdbiaadUgaaaaak8aacaGLOaGaayzkaaqcfa4dbm aalaaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbsacaWGQbGa ey4kaSIaam4AaiabgUcaRiaaiodaaaqcfa4aaeWaaOqaaKqzGeGaeq iUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaa jugibiaadQgacqGHsislcaWGRbaaaaGcbaqcfa4aaeWaaOqaaKqzGe GaeqiUdexcfa4damaaCaaaleqabaqcLbsapeGaaGOmaaaacqGHRaWk cqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaajuaGdaahaaWcbe qaaKqzGeGaamOAaiabgUcaRiaaigdaaaaaaaGcbaqcfa4aa8qCaOqa aKqzGeGaamiEaKqbaoaaCaaaleqabaqcLbsacaWGQbGaey4kaSIaam 4AaiabgUcaRiaadkhaaaqcfa4damaabmaakeaajugibiaaigdacqGH RaWkcaWG4bGaey4kaSIaamiEaKqbaoaaCaaaleqabaqcLbsacaaIYa aaaaGccaGLOaGaayzkaaqcLbsapeGaamyzaKqba+aadaahaaWcbeqa aKqzGeWdbiabgkHiTiabeI7aXLqbaoaabmaaleaajugibiaadMgacq GHRaWkcaaIXaaaliaawIcacaGLPaaajugibiaadIhaaaWdaiaadsga caWG4baal8qabaqcLbsacaaIWaaaleaajugibiabg6HiLcGaey4kIi paaaaa@B395@

=α i=0 j=0 k=0 ( 1 ) i ( α1 i ) ( i j )( j k ) ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 [ { θ( i+1 ) } 2 Γ( j+k+r+1 ) +{ θ( i+1 ) }Γ( j+k+r+2 ) +Γ( j+k+r+3 ) θ r ( i+1 ) j+k+r+3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeqySdewcfa4aaabCaOqaaKqbaoaaqahakeaajuaGdaaeWbGcbaqc fa4aaeWaaOqaaKqzGeGaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa 4aaWbaaSqabeaajugibiaadMgaaaaaleaajugibiaadUgacqGH9aqp caaIWaaaleaajugibiabg6HiLcGaeyyeIuoaaSqaaKqzGeGaamOAai abg2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5aKqbaoaabmaa keaajugibuaabeqaceaaaOqaaKqzGeaeaaaaaaaaa8qacqaHXoqycq GHsislcaaIXaaak8aabaqcLbsapeGaamyAaaaaaOWdaiaawIcacaGL PaaaaSqaaKqzGeGaamyAaiabg2da9iaaicdaaSqaaKqzGeGaeyOhIu kacqGHris5aKqbaoaabmaakeaajugibuaabeqaceaaaOqaaKqzGeWd biaadMgaaOWdaeaajugib8qacaWGQbaaaaGcpaGaayjkaiaawMcaaK qbaoaabmaakeaajugibuaabeqaceaaaOqaaKqzGeWdbiaadQgaaOWd aeaajugib8qacaWGRbaaaaGcpaGaayjkaiaawMcaaKqba+qadaWcaa Gcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGOmaaGccaGL OaGaayzkaaqcfa4aaWbaaSqabeaajugibiaadQgacqGHsislcaWGRb aaaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4damaaCaaaleqa baqcLbsapeGaaGOmaaaacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaaki aawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGaamOAaiabgUcaRiaa igdaaaaaaKqbaoaadmaakeaajuaGdaWcaaqcLbsaeaqabOqaaKqbao aacmaakeaajugibiabeI7aXLqbaoaabmaakeaajugibiaadMgacqGH RaWkcaaIXaaakiaawIcacaGLPaaaaiaawUhacaGL9baajuaGdaahaa WcbeqaaKqzGeGaaGOmaaaacqqHtoWrjuaGdaqadaGcbaqcLbsacaWG QbGaey4kaSIaam4AaiabgUcaRiaadkhacqGHRaWkcaaIXaaakiaawI cacaGLPaaaaeaajugibiabgUcaRKqbaoaacmaakeaajugibiabeI7a XLqbaoaabmaakeaajugibiaadMgacqGHRaWkcaaIXaaakiaawIcaca GLPaaaaiaawUhacaGL9baajugibiabfo5ahLqbaoaabmaakeaajugi biaadQgacqGHRaWkcaWGRbGaey4kaSIaamOCaiabgUcaRiaaikdaaO GaayjkaiaawMcaaaqaaKqzGeGaey4kaSIaeu4KdCucfa4aaeWaaOqa aKqzGeGaamOAaiabgUcaRiaadUgacqGHRaWkcaWGYbGaey4kaSIaaG 4maaGccaGLOaGaayzkaaaaaeaajugibiabeI7aXLqbaoaaCaaabeqa aKqzGeGaamOCaaaajuaGdaqadaGcbaqcLbsacaWGPbGaey4kaSIaaG ymaaGccaGLOaGaayzkaaqcfa4aaWbaaeqabaqcLbsacaWGQbGaey4k aSIaam4AaiabgUcaRiaadkhacqGHRaWkcaaIZaaaaaaaaOGaay5wai aaw2faaaaa@D45E@ (10)       

Since equation (10) is a convergent series for all r0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOCai abgwMiZkaaicdaaaa@39F1@ , all the moments or ESD exists. The first four moments about origin of ESD can thus be expressed as

μ 1 =α i=0 j=0 k=0 ( 1 ) i ( α1 i ) ( i j )( j k ) ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 [ θ 2 ( i+1 ) 2 Γ( j+k+2 ) +θ( i+1 )Γ( j+k+3 ) +Γ( j+k+4 ) θ ( i+1 ) j+k+4 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaWgaaqaaKqzadGaaGymaaWcbeaadaahaaqabeaajugWaiadacUH YaIOaaqcLbsacqGH9aqpcqaHXoqyjuaGdaaeWbGcbaqcfa4aaabCaO qaaKqbaoaaqahakeaajuaGdaqadaGcbaqcLbsacqGHsislcaaIXaaa kiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGaamyAaaaaaSqaaK qzGeGaam4Aaiabg2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5 aaWcbaqcLbsacaWGQbGaeyypa0JaaGimaaWcbaqcLbsacqGHEisPai abggHiLdqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsaqaaa aaaaaaWdbiabeg7aHjabgkHiTiaaigdaaOWdaeaajugib8qacaWGPb aaaaGcpaGaayjkaiaawMcaaaWcbaqcLbsacaWGPbGaeyypa0JaaGim aaWcbaqcLbsacqGHEisPaiabggHiLdqcfa4aaeWaaOqaaKqzGeqbae qabiqaaaGcbaqcLbsapeGaamyAaaGcpaqaaKqzGeWdbiaadQgaaaaa k8aacaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcba qcLbsapeGaamOAaaGcpaqaaKqzGeWdbiaadUgaaaaak8aacaGLOaGa ayzkaaqcfa4dbmaalaaakeaajuaGdaqadaGcbaqcLbsacqaH4oqCcq GHRaWkcaaIYaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGa amOAaiabgkHiTiaadUgaaaaakeaajuaGdaqadaGcbaqcLbsacqaH4o qCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaaaiabgUcaRiabeI7a XjabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLb sacaWGQbGaey4kaSIaaGymaaaaaaqcfa4aamWaaOqaaKqbaoaalaaa jugibqaabeqcfayaaKqzGeGaeqiUdexcfa4aaWbaaeqabaqcLbsaca aIYaaaaKqbaoaabmaabaqcLbsacaWGPbGaey4kaSIaaGymaaqcfaOa ayjkaiaawMcaamaaCaaabeqaaKqzGeGaaGOmaaaacqqHtoWrjuaGda qadaqaaKqzGeGaamOAaiabgUcaRiaadUgacqGHRaWkcaaIYaaajuaG caGLOaGaayzkaaaabaqcLbsacqGHRaWkcqaH4oqCjuaGdaqadaqaaK qzGeGaamyAaiabgUcaRiaaigdaaKqbakaawIcacaGLPaaajugibiab fo5ahLqbaoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRi aaiodaaKqbakaawIcacaGLPaaaaeaajugibiabgUcaRiabfo5ahLqb aoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRiaaisdaaK qbakaawIcacaGLPaaaaaqaaKqzGeGaeqiUdexcfa4aaeWaaeaajugi biaadMgacqGHRaWkcaaIXaaajuaGcaGLOaGaayzkaaWaaWbaaeqaba WaaWbaaeqabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRiaaisda aaaaaaaaaOGaay5waiaaw2faaaaa@D19A@

μ 2 =α i=0 j=0 k=0 ( 1 ) i ( α1 i ) ( i j )( j k ) ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 [ θ 2 ( i+1 ) 2 Γ( j+k+3 ) +θ( i+1 )Γ( j+k+4 ) +Γ( j+k+5 ) θ 2 ( i+1 ) j+k+5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaWgaaqaaKqzadGaaGOmaaWcbeaadaahaaqabeaajugWaiadacUH YaIOaaqcLbsacqGH9aqpcqaHXoqyjuaGdaaeWbGcbaqcfa4aaabCaO qaaKqbaoaaqahakeaajuaGdaqadaGcbaqcLbsacqGHsislcaaIXaaa kiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGaamyAaaaaaSqaaK qzGeGaam4Aaiabg2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5 aaWcbaqcLbsacaWGQbGaeyypa0JaaGimaaWcbaqcLbsacqGHEisPai abggHiLdqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsaqaaa aaaaaaWdbiabeg7aHjabgkHiTiaaigdaaOWdaeaajugib8qacaWGPb aaaaGcpaGaayjkaiaawMcaaaWcbaqcLbsacaWGPbGaeyypa0JaaGim aaWcbaqcLbsacqGHEisPaiabggHiLdqcfa4aaeWaaOqaaKqzGeqbae qabiqaaaGcbaqcLbsapeGaamyAaaGcpaqaaKqzGeWdbiaadQgaaaaa k8aacaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcba qcLbsapeGaamOAaaGcpaqaaKqzGeWdbiaadUgaaaaak8aacaGLOaGa ayzkaaqcfa4dbmaalaaakeaajuaGdaqadaGcbaqcLbsacqaH4oqCcq GHRaWkcaaIYaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGa amOAaiabgkHiTiaadUgaaaaakeaajuaGdaqadaGcbaqcLbsacqaH4o qCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaaaiabgUcaRiabeI7a XjabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLb sacaWGQbGaey4kaSIaaGymaaaaaaqcfa4aamWaaOqaaKqbaoaalaaa jugibqaabeqcfayaaKqzGeGaeqiUdexcfa4aaWbaaeqabaqcLbsaca aIYaaaaKqbaoaabmaabaqcLbsacaWGPbGaey4kaSIaaGymaaqcfaOa ayjkaiaawMcaamaaCaaabeqaaKqzGeGaaGOmaaaacqqHtoWrjuaGda qadaqaaKqzGeGaamOAaiabgUcaRiaadUgacqGHRaWkcaaIZaaajuaG caGLOaGaayzkaaaabaqcLbsacqGHRaWkcqaH4oqCjuaGdaqadaqaaK qzGeGaamyAaiabgUcaRiaaigdaaKqbakaawIcacaGLPaaajugibiab fo5ahLqbaoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRi aaisdaaKqbakaawIcacaGLPaaaaeaajugibiabgUcaRiabfo5ahLqb aoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRiaaiwdaaK qbakaawIcacaGLPaaaaaqaaKqzGeGaeqiUdexcfa4aaWbaaeqabaqc LbsacaaIYaaaaKqbaoaabmaabaqcLbsacaWGPbGaey4kaSIaaGymaa qcfaOaayjkaiaawMcaamaaCaaabeqaaKqzGeGaamOAaiabgUcaRiaa dUgacqGHRaWkcaaI1aaaaaaaaOGaay5waiaaw2faaaaa@D378@

μ 3 =α i=0 j=0 k=0 ( 1 ) i ( α1 i ) ( i j )( j k ) ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 [ θ 2 ( i+1 ) 2 Γ( j+k+4 ) +θ( i+1 )Γ( j+k+5 ) +Γ( j+k+6 ) θ 3 ( i+1 ) j+k+6 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaWgaaqaaKqzadGaaG4maaWcbeaadaahaaqabeaajugWaiadacUH YaIOaaqcLbsacqGH9aqpcqaHXoqyjuaGdaaeWbGcbaqcfa4aaabCaO qaaKqbaoaaqahakeaajuaGdaqadaGcbaqcLbsacqGHsislcaaIXaaa kiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGaamyAaaaaaSqaaK qzGeGaam4Aaiabg2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5 aaWcbaqcLbsacaWGQbGaeyypa0JaaGimaaWcbaqcLbsacqGHEisPai abggHiLdqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsaqaaa aaaaaaWdbiabeg7aHjabgkHiTiaaigdaaOWdaeaajugib8qacaWGPb aaaaGcpaGaayjkaiaawMcaaaWcbaqcLbsacaWGPbGaeyypa0JaaGim aaWcbaqcLbsacqGHEisPaiabggHiLdqcfa4aaeWaaOqaaKqzGeqbae qabiqaaaGcbaqcLbsapeGaamyAaaGcpaqaaKqzGeWdbiaadQgaaaaa k8aacaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcba qcLbsapeGaamOAaaGcpaqaaKqzGeWdbiaadUgaaaaak8aacaGLOaGa ayzkaaqcfa4dbmaalaaakeaajuaGdaqadaGcbaqcLbsacqaH4oqCcq GHRaWkcaaIYaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGa amOAaiabgkHiTiaadUgaaaaakeaajuaGdaqadaGcbaqcLbsacqaH4o qCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaaaiabgUcaRiabeI7a XjabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLb sacaWGQbGaey4kaSIaaGymaaaaaaqcfa4aamWaaOqaaKqbaoaalaaa jugibqaabeqcfayaaKqzGeGaeqiUdexcfa4aaWbaaeqabaqcLbsaca aIYaaaaKqbaoaabmaabaqcLbsacaWGPbGaey4kaSIaaGymaaqcfaOa ayjkaiaawMcaamaaCaaabeqaaKqzGeGaaGOmaaaacqqHtoWrjuaGda qadaqaaKqzGeGaamOAaiabgUcaRiaadUgacqGHRaWkcaaI0aaajuaG caGLOaGaayzkaaaabaqcLbsacqGHRaWkcqaH4oqCjuaGdaqadaqaaK qzGeGaamyAaiabgUcaRiaaigdaaKqbakaawIcacaGLPaaajugibiab fo5ahLqbaoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRi aaiwdaaKqbakaawIcacaGLPaaaaeaajugibiabgUcaRiabfo5ahLqb aoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRiaaiAdaaK qbakaawIcacaGLPaaaaaqaaKqzGeGaeqiUdexcfa4aaWbaaeqabaqc LbsacaaIZaaaaKqbaoaabmaabaqcLbsacaWGPbGaey4kaSIaaGymaa qcfaOaayjkaiaawMcaamaaCaaabeqaaKqzGeGaamOAaiabgUcaRiaa dUgacqGHRaWkcaaI2aaaaaaaaOGaay5waiaaw2faaaaa@D37E@

μ 4 =α i=0 j=0 k=0 ( 1 ) i ( α1 i ) ( i j )( j k ) ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 [ θ 2 ( i+1 ) 2 Γ( j+k+5 ) +θ( i+1 )Γ( j+k+6 ) +Γ( j+k+7 ) θ 4 ( i+1 ) j+k+7 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaWgaaqaaKqzadGaaGinaaWcbeaadaahaaqabeaajugWaiadacUH YaIOaaqcLbsacqGH9aqpcqaHXoqyjuaGdaaeWbGcbaqcfa4aaabCaO qaaKqbaoaaqahakeaajuaGdaqadaGcbaqcLbsacqGHsislcaaIXaaa kiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGaamyAaaaaaSqaaK qzGeGaam4Aaiabg2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5 aaWcbaqcLbsacaWGQbGaeyypa0JaaGimaaWcbaqcLbsacqGHEisPai abggHiLdqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsaqaaa aaaaaaWdbiabeg7aHjabgkHiTiaaigdaaOWdaeaajugib8qacaWGPb aaaaGcpaGaayjkaiaawMcaaaWcbaqcLbsacaWGPbGaeyypa0JaaGim aaWcbaqcLbsacqGHEisPaiabggHiLdqcfa4aaeWaaOqaaKqzGeqbae qabiqaaaGcbaqcLbsapeGaamyAaaGcpaqaaKqzGeWdbiaadQgaaaaa k8aacaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcba qcLbsapeGaamOAaaGcpaqaaKqzGeWdbiaadUgaaaaak8aacaGLOaGa ayzkaaqcfa4dbmaalaaakeaajuaGdaqadaGcbaqcLbsacqaH4oqCcq GHRaWkcaaIYaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGa amOAaiabgkHiTiaadUgaaaaakeaajuaGdaqadaGcbaqcLbsacqaH4o qCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaaaiabgUcaRiabeI7a XjabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLb sacaWGQbGaey4kaSIaaGymaaaaaaqcfa4aamWaaOqaaKqbaoaalaaa jugibqaabeqcfayaaKqzGeGaeqiUdexcfa4aaWbaaeqabaqcLbsaca aIYaaaaKqbaoaabmaabaqcLbsacaWGPbGaey4kaSIaaGymaaqcfaOa ayjkaiaawMcaamaaCaaabeqaaKqzGeGaaGOmaaaacqqHtoWrjuaGda qadaqaaKqzGeGaamOAaiabgUcaRiaadUgacqGHRaWkcaaI1aaajuaG caGLOaGaayzkaaaabaqcLbsacqGHRaWkcqaH4oqCjuaGdaqadaqaaK qzGeGaamyAaiabgUcaRiaaigdaaKqbakaawIcacaGLPaaajugibiab fo5ahLqbaoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRi aaiAdaaKqbakaawIcacaGLPaaaaeaajugibiabgUcaRiabfo5ahLqb aoaabmaabaqcLbsacaWGQbGaey4kaSIaam4AaiabgUcaRiaaiEdaaK qbakaawIcacaGLPaaaaaqaaKqzGeGaeqiUdexcfa4aaWbaaeqabaqc LbsacaaI0aaaaKqbaoaabmaabaqcLbsacaWGPbGaey4kaSIaaGymaa qcfaOaayjkaiaawMcaamaaCaaabeqaaKqzGeGaamOAaiabgUcaRiaa dUgacqGHRaWkcaaI3aaaaaaaaOGaay5waiaaw2faaaaa@D384@

Therefore, variance of ESD can thus be obtained using the formula

μ 2 = μ 2 ( μ 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 wcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabg2da9iabeY7a TTWaaSbaaeaajugWaiaaikdaaSqabaWaaWbaaeqabaqcLbmacWaGGB OmGikaaKqzGeGaeyOeI0scfa4aaeWaaOqaaKqzGeGaeqiVd02cdaWg aaqaaKqzadGaaGymaaWcbeaadaahaaqabeaajugWaiadacUHYaIOaa aakiaawIcacaGLPaaalmaaCaaabeqaaKqzadGaaGOmaaaaaaa@52DE@ .

Harmonic mean

Harmonic mean of ESD can be obtained as

E( 1 X )= 0 1 x g(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyraK qbaoaabmaabaWaaSaaaeaajugibiaaigdaaKqbagaajugibiaadIfa aaaajuaGcaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWdXbGcbaqcfa 4aaSaaaeaajugibiaaigdaaKqbagaajugibiaadIhaaaGaam4zaiaa cIcacaWG4bGaaiykaiaadsgacaWG4baaleaajugibiaaicdaaSqaaK qzGeGaeyOhIukacqGHRiI8aaaa@4E44@

=α i=0 j=0 k=0 ( 1 ) i ( α1 i ) ( i j )( j k ) θ j+k+3 ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 [ { θ( i+1 ) } 2 Γ( j+k ) +{ θ( i+1 ) }Γ( j+k+1 ) +Γ( j+k+2 ) { θ( i+1 ) } j+k+2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeqySdewcfa4aaabCaOqaaKqbaoaaqahakeaajuaGdaaeWbGcbaqc fa4aaeWaaOqaaKqzGeGaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa 4aaWbaaSqabeaajugibiaadMgaaaaaleaajugibiaadUgacqGH9aqp caaIWaaaleaajugibiabg6HiLcGaeyyeIuoaaSqaaKqzGeGaamOAai abg2da9iaaicdaaSqaaKqzGeGaeyOhIukacqGHris5aKqbaoaabmaa keaajugibuaabeqaceaaaOqaaKqzGeaeaaaaaaaaa8qacqaHXoqycq GHsislcaaIXaaak8aabaqcLbsapeGaamyAaaaaaOWdaiaawIcacaGL PaaaaSqaaKqzGeGaamyAaiabg2da9iaaicdaaSqaaKqzGeGaeyOhIu kacqGHris5aKqbaoaabmaakeaajugibuaabeqaceaaaOqaaKqzGeWd biaadMgaaOWdaeaajugib8qacaWGQbaaaaGcpaGaayjkaiaawMcaaK qbaoaabmaakeaajugibuaabeqaceaaaOqaaKqzGeWdbiaadQgaaOWd aeaajugib8qacaWGRbaaaaGcpaGaayjkaiaawMcaaKqba+qadaWcaa GcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaamOAaiabgUca RiaadUgacqGHRaWkcaaIZaaaaKqbaoaabmaakeaajugibiabeI7aXj abgUcaRiaaikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLbsa caWGQbGaeyOeI0Iaam4AaaaaaOqaaKqbaoaabmaakeaajugibiabeI 7aXLqba+aadaahaaWcbeqaaKqzGeWdbiaaikdaaaGaey4kaSIaeqiU deNaey4kaSIaaGOmaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaaju gibiaadQgacqGHRaWkcaaIXaaaaaaajuaGdaWadaGcbaqcfa4aaSaa aKqzGeabaeqakeaajuaGdaGadaGcbaqcLbsacqaH4oqCjuaGdaqada GcbaqcLbsacaWGPbGaey4kaSIaaGymaaGccaGLOaGaayzkaaaacaGL 7bGaayzFaaqcfa4aaWbaaSqabeaajugibiaaikdaaaGaeu4KdCucfa 4aaeWaaOqaaKqzGeGaamOAaiabgUcaRiaadUgaaOGaayjkaiaawMca aaqaaKqzGeGaey4kaSscfa4aaiWaaOqaaKqzGeGaeqiUdexcfa4aae WaaOqaaKqzGeGaamyAaiabgUcaRiaaigdaaOGaayjkaiaawMcaaaGa ay5Eaiaaw2haaKqzGeGaeu4KdCucfa4aaeWaaOqaaKqzGeGaamOAai abgUcaRiaadUgacqGHRaWkcaaIXaaakiaawIcacaGLPaaaaeaajugi biabgUcaRiabfo5ahLqbaoaabmaakeaajugibiaadQgacqGHRaWkca WGRbGaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaeaajuaGdaGadaGc baqcLbsacqaH4oqCjuaGdaqadaGcbaqcLbsacaWGPbGaey4kaSIaaG ymaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaqcfa4aaWbaaSqabeaa jugibiaadQgacqGHRaWkcaWGRbGaey4kaSIaaGOmaaaaaaaakiaawU facaGLDbaaaaa@D3E7@ (11)            

Moment generating function

Moment generating function of ESD can be obtained as

M x (t)=E[ e tX ]= 0 e tx g(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaK qbaoaaBaaaleaajugibiaadIhaaSqabaqcLbsacaGGOaGaamiDaiaa cMcacqGH9aqpcaWGfbqcfa4aamWaaOqaaKqzGeGaamyzaKqbaoaaCa aaleqabaqcLbsacaWG0bGaaGPaVlaadIfaaaaakiaawUfacaGLDbaa jugibiabg2da9KqbaoaapehakeaajugibiaadwgajuaGdaahaaWcbe qaaKqzGeGaamiDaiaaykW7caWG4baaaiaadEgacaGGOaGaamiEaiaa cMcacaWGKbGaamiEaaWcbaqcLbsacaaIWaaaleaajugibiabg6HiLc Gaey4kIipaaaa@5A4F@

Using Taylor’s series expansion, we get

M X (t)= 0 ( 1+tX ( tX ) 2 2! +... )g(x)dx = 0 r=0 t r r! x r g(x)dx = 0 r=0 t r r! μ r ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGnbqcfa4aaSbaaSqaaKqzGeGaamiwaaWcbeaajugibiaacIcacaWG 0bGaaiykaiabg2da9KqbaoaapehakeaajuaGdaqadaGcbaqcLbsaca aIXaGaey4kaSIaamiDaiaaykW7caWGybqcfa4aaSaaaOqaaKqbaoaa bmaakeaajugibiaadshacaaMc8UaamiwaaGccaGLOaGaayzkaaqcfa 4aaWbaaSqabeaajugibiaaikdaaaaakeaajugibiaaikdacaGGHaaa aiabgUcaRiaac6cacaGGUaGaaiOlaaGccaGLOaGaayzkaaqcLbsaca WGNbGaaiikaiaadIhacaGGPaGaamizaiaadIhaaSqaaKqzGeGaaGim aaWcbaqcLbsacqGHEisPaiabgUIiYdaakeaajugibiabg2da9Kqbao aapehakeaajuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGeGaamiDaKqb aoaaCaaaleqabaqcLbmacaWGYbaaaaGcbaqcLbsacaWGYbqcLbmaca GGHaaaaKqzGeGaamiEaSWaaWbaaeqabaqcLbmacaWGYbaaaKqzGeGa am4zaiaacIcacaWG4bGaaiykaiaadsgacaWG4baaleaajugibiaadk hacqGH9aqpcaaIWaaaleaajugibiabg6HiLcGaeyyeIuoaaSqaaKqz GeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYdGaeyypa0tcfa4aa8 qCaOqaaKqbaoaaqahakeaajuaGdaWcaaGcbaqcLbsacaWG0bWcdaah aaqabeaajugWaiaadkhaaaaakeaajugibiaadkhajugWaiaacgcaaa qcLbsacqaH8oqBlmaaDaaabaqcLbmacaWGYbaaleaajugWaiaacEca aaaaleaajugibiaadkhacqGH9aqpcaaIWaaaleaajugibiabg6HiLc GaeyyeIuoaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIi Ydaaaaa@9EB6@

=α i=0 j=0 k=0 r=0 ( 1 ) i ( α1 i )( i j )( j k ) t r r! ( θ+2 ) jk ( θ 2 +θ+2 ) j+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeqySdewcfa4aaabCaOqaaKqbaoaaqahakeaajuaGdaaeWbGcbaqc fa4aaabCaOqaaKqbaoaabmaakeaajugibiabgkHiTiaaigdaaOGaay jkaiaawMcaaaWcbaqcLbsacaWGYbGaeyypa0JaaGimaaWcbaqcLbsa cqGHEisPaiabggHiLdaaleaajugibiaadUgacqGH9aqpcaaIWaaale aajugibiabg6HiLcGaeyyeIuoaaSqaaKqzGeGaamOAaiabg2da9iaa icdaaSqaaKqzGeGaeyOhIukacqGHris5aaWcbaqcLbsacaWGPbGaey ypa0JaaGimaaWcbaqcLbsacqGHEisPaiabggHiLdqcfa4aaWbaaSqa beaajugibiaadMgaaaqcfaieaaaaaaaaa8qadaqadaGcpaqaaKqzGe qbaeqabiqaaaGcbaqcLbsapeGaeqySdeMaeyOeI0IaaGymaaGcpaqa aKqzGeWdbiaadMgaaaaakiaawIcacaGLPaaajuaGdaqadaGcpaqaaK qzGeqbaeqabiqaaaGcbaqcLbsacaWGPbaakeaajugibiaadQgaaaaa k8qacaGLOaGaayzkaaqcfa4aaeWaaOWdaeaajugibuaabeqaceaaaO qaaKqzGeGaamOAaaGcbaqcLbsacaWGRbaaaaGcpeGaayjkaiaawMca aKqbaoaalaaakeaajugibiaadshajuaGdaahaaWcbeqaaKqzGeGaam OCaaaaaOqaaKqzGeGaamOCaiaacgcaaaqcfa4aaSaaaOqaaKqbaoaa bmaakeaajugibiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaK qbaoaaCaaabeqaaKqzGeGaamOAaiabgkHiTiaadUgaaaaakeaajuaG daqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOmaa aacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaajuaG daahaaWcbeqaaKqzGeGaamOAaiabgUcaRiaaigdaaaaaaaaa@9484@

×[ θ 2 ( i+1 ) 2 Γ( j+k+r+1 ) +θ( i+1 )Γ( j+k+r+2 ) +Γ( j+k+r+2 ) θ r ( i+1 ) j+k+r+3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGHxdaTjuaGdaWadaGcbaqcfa4aaSaaaKqzGeabaeqakeaa jugibiabeI7aXLqbaoaaCaaabeqaaKqzGeGaaGOmaaaajuaGdaqada GcbaqcLbsacaWGPbGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4a aWbaaeqabaqcLbsacaaIYaaaaiabfo5ahLqbaoaabmaakeaajugibi aadQgacqGHRaWkcaWGRbGaey4kaSIaamOCaiabgUcaRiaaigdaaOGa ayjkaiaawMcaaaqaaKqzGeGaey4kaSIaeqiUdexcfa4aaeWaaOqaaK qzGeGaamyAaiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqzGeGaeu4K dCucfa4aaeWaaOqaaKqzGeGaamOAaiabgUcaRiaadUgacqGHRaWkca WGYbGaey4kaSIaaGOmaaGccaGLOaGaayzkaaaabaqcLbsacqGHRaWk cqqHtoWrjuaGdaqadaGcbaqcLbsacaWGQbGaey4kaSIaam4AaiabgU caRiaadkhacqGHRaWkcaaIYaaakiaawIcacaGLPaaaaaqaaKqzGeGa eqiUdexcfa4aaWbaaeqabaqcLbsacaWGYbaaaKqbaoaabmaakeaaju gibiaadMgacqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaahaaWc beqaaKqzGeGaamOAaiabgUcaRiaadUgacqGHRaWkcaWGYbGaey4kaS IaaG4maaaaaaaakiaawUfacaGLDbaaaaa@82C0@ (12)

Order statistics of ESD

The pdf of the r th order statistics Y r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamywaS WaaSbaaeaajugWaiaadkhaaSqabaaaaa@39B4@ of ESD can be obtained a

g Y r ( x )= n! ( r1 )!(nr)! g(x) [ G(x) ] r1 [ 1G(x) ] nr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaK qbaoaaBaaaleaajugibiaadMfajuaGdaWgaaadbaqcLbmacaWGYbaa meqaaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPa aajugibiabg2da9Kqbaoaalaaakeaajugibiaad6gacaGGHaaakeaa juaGdaqadaGcbaqcLbsacaWGYbGaeyOeI0IaaGymaaGccaGLOaGaay zkaaqcLbsacaGGHaGaaiikaiaad6gacqGHsislcaWGYbGaaiykaiaa cgcaaaGaam4zaiaacIcacaWG4bGaaiykaKqbaoaadmaakeaajugibi aadEeacaGGOaGaamiEaiaacMcaaOGaay5waiaaw2faaSWaaWbaaeqa baqcLbmacaWGYbGaeyOeI0IaaGymaaaajuaGdaWadaGcbaqcLbsaca aIXaGaeyOeI0Iaam4raiaacIcacaWG4bGaaiykaaGccaGLBbGaayzx aaWcdaahaaqabeaajugWaiaad6gacqGHsislcaWGYbaaaaaa@69AB@

= n! ( r1 )!(nr)! [ α θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx { 1( 1+ θx( θx+θ+2 ) θ 2 +θ+2 ) e θx } α1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaKGbagaajugibiaad6gacaGGHaaajyaGbaqcfa4aaeWa aKGbagaajugibiaadkhacqGHsislcaaIXaaajyaGcaGLOaGaayzkaa qcLbsacaGGHaGaaiikaiaad6gacqGHsislcaWGYbGaaiykaiaacgca aaqcfa4aamWaaKGbagaajuaGqaaaaaaaaaWdbmaalaaajyaGpaqaaK qzGeWdbiabeg7aHjabeI7aXLqba+aadaahaaqcgayabeaajugib8qa caaIZaaaaaqcga4daeaajugib8qacqaH4oqCjuaGpaWaaWbaaKGbag qabaqcLbsapeGaaGOmaaaacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaa aKqbaoaabmaajyaGpaqaaKqzGeWdbiaaigdacqGHRaWkcaWG4bGaey 4kaSIaamiEaKqba+aadaahaaqcgayabeaajugib8qacaaIYaaaaaqc gaOaayjkaiaawMcaaKqzGeGaamyzaKqba+aadaahaaqcgayabeaaju gib8qacqGHsislcqaH4oqCcaWG4baaaKqba+aadaGadaqcgayaaKqz GeWdbiaaigdacqGHsisljuaGdaqadaqcgayaaKqzGeGaaGymaiabgU caRKqbaoaalaaajyaGpaqaaKqzGeWdbiabeI7aXjaadIhajuaGdaqa daqcga4daeaajugib8qacqaH4oqCcaWG4bGaey4kaSIaeqiUdeNaey 4kaSIaaGOmaaqcgaOaayjkaiaawMcaaaWdaeaajugib8qacqaH4oqC juaGpaWaaWbaaKGbagqabaqcLbsapeGaaGOmaaaacqGHRaWkcqaH4o qCcqGHRaWkcaaIYaaaaaqcgaOaayjkaiaawMcaaKqzGeGaamyzaKqb a+aadaahaaqcgayabeaajugib8qacqGHsislcqaH4oqCcaWG4baaaa qcga4daiaawUhacaGL9baajuaGdaahaaqcgayabeaajugib8qacqaH XoqycqGHsislcaaIXaaaaaqcga4daiaawUfacaGLDbaaaaa@9FBE@

× [ 1{ 1+ θx( θx+θ+2 ) θ 2 +θ+2 } e θx ] α(r1) [ 1 { 1( 1+ θx( θx+θ+2 ) θ 2 +θ+2 ) e θx } α ] nr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaey41aq Bcfa4aamWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaacmaakeaajugi biaaigdacqGHRaWkjuaGqaaaaaaaaaWdbmaalaaak8aabaqcLbsape GaeqiUdeNaamiEaKqbaoaabmaak8aabaqcLbsapeGaeqiUdeNaamiE aiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaWdae aajugib8qacqaH4oqCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaa aiabgUcaRiabeI7aXjabgUcaRiaaikdaaaaak8aacaGL7bGaayzFaa qcLbsapeGaamyzaKqba+aadaahaaWcbeqaaKqzGeWdbiabgkHiTiab eI7aXjaadIhaaaaak8aacaGLBbGaayzxaaqcfa4aaWbaaSqabeaaju gib8qacqaHXoqycaGGOaGaamOCaiabgkHiTiaaigdacaGGPaaaaKqb a+aadaWadaGcbaqcLbsacaaIXaGaeyOeI0scfa4aaiWaaOqaaKqzGe GaaGymaiabgkHiTKqbaoaabmaakeaajugibiaaigdacqGHRaWkjuaG peWaaSaaaOWdaeaajugib8qacqaH4oqCcaWG4bqcfa4aaeWaaOWdae aajugib8qacqaH4oqCcaWG4bGaey4kaSIaeqiUdeNaey4kaSIaaGOm aaGccaGLOaGaayzkaaaapaqaaKqzGeWdbiabeI7aXLqba+aadaahaa WcbeqaaKqzGeWdbiaaikdaaaGaey4kaSIaeqiUdeNaey4kaSIaaGOm aaaaaOWdaiaawIcacaGLPaaajugib8qacaWGLbqcfa4damaaCaaale qabaqcLbsapeGaeyOeI0IaeqiUdeNaamiEaaaaaOWdaiaawUhacaGL 9baajuaGdaahaaWcbeqaaKqzGeGaeqySdegaaaGccaGLBbGaayzxaa qcfa4aaWbaaSqabeaajugibiaad6gacqGHsislcaWGYbaaaaaa@9935@                      (13)     

The pdf of first order Statistics Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamywaS WaaSbaaeaajugWaiaaigdaaSqabaaaaa@3978@ of ESD can be obtained as

g Y 1 (x)=n [ 1G(x) ] n1 g(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaK qbaoaaBaaaleaajugibiaadMfalmaaBaaameaajugWaiaaigdaaWqa baaaleqaaKqzGeGaaiikaiaadIhacaGGPaGaeyypa0JaamOBaKqbao aadmaakeaajugibiaaigdacqGHsislcaWGhbGaaiikaiaadIhacaGG PaaakiaawUfacaGLDbaalmaaCaaabeqaaKqzadGaamOBaiabgkHiTi aaigdaaaqcLbsacaWGNbGaaiikaiaadIhacaGGPaaaaa@5057@

=n [ 1 { 1( 1+ θx( θx+θ+2 ) θ 2 +θ+2 ) e θx } α ] n1 α θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaamOBaKqbaoaadmaakeaajugibiaaigdacqGHsisljuaGdaGadaGc baqcLbsacaaIXaGaeyOeI0scfa4aaeWaaOqaaKqzGeaeaaaaaaaaa8 qacaaIXaGaey4kaSscfa4aaSaaaOWdaeaajugib8qacqaH4oqCcaWG 4bqcfa4aaeWaaOWdaeaajugib8qacqaH4oqCcaWG4bGaey4kaSIaeq iUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaapaqaaKqzGeWdbiab eI7aXLqba+aadaahaaWcbeqaaKqzGeWdbiaaikdaaaGaey4kaSIaeq iUdeNaey4kaSIaaGOmaaaaaOWdaiaawIcacaGLPaaajugib8qacaWG Lbqcfa4damaaCaaaleqabaqcLbsapeGaeyOeI0IaeqiUdeNaamiEaa aaaOWdaiaawUhacaGL9baajuaGdaahaaWcbeqaaKqzGeWdbiabeg7a HbaaaOWdaiaawUfacaGLDbaalmaaCaaabeqaaKqzadGaamOBaiabgk HiTiaaigdaaaqcfa4dbmaalaaak8aabaqcLbsapeGaeqySdeMaeqiU dexcfa4damaaCaaaleqabaqcLbsapeGaaG4maaaaaOWdaeaajugib8 qacqaH4oqCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaaaiabgUca RiabeI7aXjabgUcaRiaaikdaaaqcfa4aaeWaaOWdaeaajugib8qaca aIXaGaey4kaSIaamiEaiabgUcaRiaadIhajuaGpaWaaWbaaSqabeaa jugib8qacaaIYaaaaaGccaGLOaGaayzkaaqcLbsacaWGLbqcfa4dam aaCaaaleqabaqcLbsapeGaeyOeI0IaeqiUdeNaamiEaaaaaaa@8A9A@

× [ 1{ 1+ θx( θx+θ+2 ) θ 2 +θ+2 } e θx ] α1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaey41aq Bcfa4aamWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaacmaakeaajugi babaaaaaaaaapeGaaGymaiabgUcaRKqbaoaalaaak8aabaqcLbsape GaeqiUdeNaamiEaKqbaoaabmaak8aabaqcLbsapeGaeqiUdeNaamiE aiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaWdae aajugib8qacqaH4oqCjuaGpaWaaWbaaSqabeaajugib8qacaaIYaaa aiabgUcaRiabeI7aXjabgUcaRiaaikdaaaaak8aacaGL7bGaayzFaa qcLbsapeGaamyzaKqba+aadaahaaWcbeqaaKqzadWdbiabgkHiTiab eI7aXjaadIhaaaaak8aacaGLBbGaayzxaaqcfa4aaWbaaSqabeaaju gWa8qacqaHXoqycqGHsislcaaIXaaaaaaa@6594@                            (14)               

The pdf of n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOBaa aa@376D@ th order statistics Y n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamywaS WaaSbaaeaajugWaiaad6gaaSqabaaaaa@39B0@ of ESD can be obtained as

g Y n (x)=n [ G(x) ] n1 g(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaS WaaSbaaeaajugWaiaadMfalmaaBaaameaajugWaiaad6gaaWqabaaa leqaaKqzGeGaaiikaiaadIhacaGGPaGaeyypa0JaamOBaKqbaoaadm aakeaajugibiaadEeacaGGOaGaamiEaiaacMcaaOGaay5waiaaw2fa aSWaaWbaaeqabaqcLbmacaWGUbGaeyOeI0IaaGymaaaajugibiaadE gacaGGOaGaamiEaiaacMcaaaa@4EF8@

= α θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx [ 1{ 1+ θx( θx+θ+2 ) θ 2 +θ+2 } e θx ] α1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfaieaaaaaaaaa8qadaWcaaGcpaqaaKqzGeWdbiabeg7aHjabeI7a XLqba+aadaahaaWcbeqaaKqzGeWdbiaaiodaaaaak8aabaqcLbsape GaeqiUdexcfa4damaaCaaaleqabaqcLbsapeGaaGOmaaaacqGHRaWk cqaH4oqCcqGHRaWkcaaIYaaaaKqbaoaabmaak8aabaqcLbsapeGaaG ymaiabgUcaRiaadIhacqGHRaWkcaWG4bqcfa4damaaCaaaleqabaqc LbsapeGaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGaamyzaKqba+aada ahaaWcbeqaaKqzadWdbiabgkHiTiabeI7aXjaadIhaaaqcfa4damaa dmaakeaajugib8qacaaIXaGaeyOeI0scfa4aaiWaaOqaaKqzGeGaaG ymaiabgUcaRKqbaoaalaaak8aabaqcLbsapeGaeqiUdeNaamiEaKqb aoaabmaak8aabaqcLbsapeGaeqiUdeNaamiEaiabgUcaRiabeI7aXj abgUcaRiaaikdaaOGaayjkaiaawMcaaaWdaeaajugib8qacqaH4oqC juaGpaWaaWbaaSqabeaajugib8qacaaIYaaaaiabgUcaRiabeI7aXj abgUcaRiaaikdaaaaakiaawUhacaGL9baajugibiaadwgal8aadaah aaqabeaajugWa8qacqGHsislcqaH4oqCcaWG4baaaaGcpaGaay5wai aaw2faaKqbaoaaCaaaleqabaqcLbmapeGaeqySdeMaeyOeI0IaaGym aaaaaaa@8427@

× [ { 1( 1+ θx( θx+θ+2 ) θ 2 +θ+2 ) e θx } α ] n1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaey41aq Bcfa4aamWaaOqaaKqbaoaacmaakeaajugibiaaigdacqGHsisljuaG daqadaGcbaqcLbsaqaaaaaaaaaWdbiaaigdacqGHRaWkjuaGdaWcaa GcpaqaaKqzGeWdbiabeI7aXjaadIhajuaGdaqadaGcpaqaaKqzGeWd biabeI7aXjaadIhacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawI cacaGLPaaaa8aabaqcLbsapeGaeqiUdexcfa4damaaCaaaleqabaqc LbsapeGaaGOmaaaacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaaaaGcpa GaayjkaiaawMcaaKqzGeWdbiaadwgajuaGpaWaaWbaaSqabeaajugW a8qacqGHsislcqaH4oqCcaWG4baaaaGcpaGaay5Eaiaaw2haaSWaaW baaeqabaqcLbmapeGaeqySdegaaaGcpaGaay5waiaaw2faaSWaaWba aeqabaqcLbmacaWGUbGaeyOeI0IaaGymaaaaaaa@698E@ (15)                                                                           

Renyi entropy of ESD

Let X be a continuous random variable following ESD. The Renyi entropy generally deals with the measures of uncertainty or spread of X. The entropy of a random variable X defined by Renyi20 can be obtained as

R E = 1 1β log( 0 [ g(x) ] β dx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaS WaaSbaaeaajugWaiaadweaaSqabaqcLbsacqGH9aqpjuaGdaWcaaGc baqcLbsacaaIXaaakeaajugibiaaigdacqGHsislcqaHYoGyaaGaci iBaiaac+gacaGGNbqcfa4aaeWaaOqaaKqbaoaapehakeaajuaGdaWa daGcbaqcLbsacaWGNbGaaiikaiaadIhacaGGPaaakiaawUfacaGLDb aajuaGdaahaaWcbeqaaKqzGeGaeqOSdigaaiaadsgacaWG4baaleaa jugibiaaicdaaSqaaKqzGeGaeyOhIukacqGHRiI8aaGccaGLOaGaay zkaaaaaa@5762@

= 1 1β [ log{ ( α θ 3 θ 2 +θ+2 ) β i=0 j=0 k=0 m=0 r=0 s=0 ( 1 ) i ( β( α1 ) i )( i j )( j k ) ( k m )( β r )( r s ) 2 km ( θ 2 +θ+2 ) j { Γ( 2j+k+m+r+s+1 ) θ r+s+1 ( β+i ) 2j+k+m+r+s+1 } } ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIXaGaeyOeI0Ia eqOSdigaaKqbaoaadmaakeaajugibiGacYgacaGGVbGaai4zaKqbao aacmaajugibqaabeGcbaqcfa4aaeWaaOqaaKqbaoaalaaakeaajugi biabeg7aHjabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaaGcba qcLbsacqaH4oqClmaaCaaabeqaaKqzadGaaGOmaaaajugibiabgUca RiabeI7aXjabgUcaRiaaikdaaaaakiaawIcacaGLPaaajuaGdaahaa WcbeqaaKqzadGaeqOSdigaaKqbaoaaqahakeaajuaGdaaeWbGcbaqc fa4aaabCaOqaaKqbaoaaqahakeaajuaGdaaeWbGcbaqcfa4aaabCaO qaaKqbaoaabmaakeaajugibiabgkHiTiaaigdaaOGaayjkaiaawMca aKqbaoaaCaaaleqabaqcLbmacaWGPbaaaKqbacbaaaaaaaaapeWaae WaaOWdaeaajugibuaabeqaceaaaOqaaKqzGeGaeqOSdiwcfa4aaeWa aOqaaKqzGeGaeqySdeMaeyOeI0IaaGymaaGccaGLOaGaayzkaaaaba qcLbsapeGaamyAaaaaaOGaayjkaiaawMcaaKqbaoaabmaak8aabaqc LbsafaqabeGabaaakeaajugibiaadMgaaOqaaKqzGeWdbiaadQgaaa aakiaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeqbaeqabiqaaaGc baqcLbsacaWGQbaakeaajugibiaadUgaaaaak8qacaGLOaGaayzkaa aal8aabaqcLbmacaWGZbGaeyypa0JaaGimaaWcbaqcLbmacqGHEisP aKqzGeGaeyyeIuoaaSqaaKqzadGaamOCaiabg2da9iaaicdaaSqaaK qzadGaeyOhIukajugibiabggHiLdaaleaajugWaiaad2gacqGH9aqp caaIWaaaleaajugWaiabg6HiLcqcLbsacqGHris5aaWcbaqcLbmaca WGRbGaeyypa0JaaGimaaWcbaqcLbmacqGHEisPaKqzGeGaeyyeIuoa aSqaaKqzadGaamOAaiabg2da9iaaicdaaSqaaKqzadGaeyOhIukaju gibiabggHiLdaaleaajugWaiaadMgacqGH9aqpcaaIWaaaleaajugW aiabg6HiLcqcLbsacqGHris5aaGcbaqcfa4dbmaabmaak8aabaqcLb safaqabeGabaaakeaajugibiaadUgaaOqaaKqzGeGaamyBaaaaaOWd biaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeqbaeqabiqaaaGcba qcLbsacqaHYoGyaOqaaKqzGeWdbiaadkhaaaaakiaawIcacaGLPaaa juaGdaqadaGcpaqaaKqzGeqbaeqabiqaaaGcbaqcLbsacaWGYbaake aajugib8qacaWGZbaaaaGccaGLOaGaayzkaaaabaqcfa4aaSaaaOqa aKqzGeGaaGOmaKqbaoaaCaaaleqabaqcLbmacaWGRbGaeyOeI0Iaam yBaaaaaOqaaKqba+aadaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWc beqaaKqzGeGaaGOmaaaacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaaki aawIcacaGLPaaalmaaCaaabeqaaKqzadGaamOAaaaaaaqcfa4dbmaa cmaakeaajuaGdaWcaaGcbaqcLbsacqqHtoWrjuaGdaqadaGcbaqcLb sacaaIYaGaamOAaiabgUcaRiaadUgacqGHRaWkcaWGTbGaey4kaSIa amOCaiabgUcaRiaadohacqGHRaWkcaaIXaaakiaawIcacaGLPaaaae aajugibiabeI7aXTWaaWbaaeqabaqcLbmacaWGYbGaey4kaSIaam4C aiabgUcaRiaaigdaaaqcfa4aaeWaaOqaaKqzGeGaeqOSdiMaey4kaS IaamyAaaGccaGLOaGaayzkaaWcdaahaaqabeaajugWaiaaikdacaWG QbGaey4kaSIaam4AaiabgUcaRiaad2gacqGHRaWkcaWGYbGaey4kaS Iaam4CaiabgUcaRiaaigdaaaaaaaGccaGL7bGaayzFaaaaa8aacaGL 7bGaayzFaaaacaGLBbGaayzxaaaaaa@0BAD@ (16)

Estimation of the parameters

Maximum likelihood estimation of parameters

Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaS WaaSbaaeaajugWaiaaigdaaSqabaqcLbsacaGGSaGaamiEaSWaaSba aeaajugWaiaaikdaaSqabaqcLbsacaGGSaGaaiOlaiaac6cacaGGUa GaaiilaiaadIhalmaaBaaabaqcLbmacaWGUbaaleqaaaaa@454E@ be the random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOBaa aa@376D@ from ESD. The likelihood function can be written as

L( α,θ )= ( α θ 3 θ 2 +θ+2 ) n i=1 n [ ( 1+x+ x 2 ) e θx { 1( 1+ θx(θx+θ+2) θ 2 +θ+2 ) e θx } α1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaK qbaoaabmaakeaajugibiabeg7aHjaacYcacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaeWaaOqaaKqbaoaalaaakeaajugibi abeg7aHjabeI7aXLqbaoaaCaaaleqabaqcLbsacaaIZaaaaaGcbaqc LbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcq aH4oqCcqGHRaWkcaaIYaaaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqa beaajugibiaad6gaaaqcfa4aaebCaOqaaKqbaoaadmaakeaajuaGda qadaGcbaqcLbsacaaIXaGaey4kaSIaamiEaiabgUcaRiaadIhajuaG daahaaWcbeqaaKqzGeGaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGaam yzaKqbaoaaCaaaleqabaqcLbsacqGHsislcqaH4oqCcaWG4baaaKqb aoaacmaakeaajugibiaaigdacqGHsisljuaGdaqadaGcbaqcLbsaca aIXaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqiUdeNaamiEaiaacIca cqaH4oqCcaWG4bGaey4kaSIaeqiUdeNaey4kaSIaaGOmaiaacMcaaO qaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikdaaaGaey4k aSIaeqiUdeNaey4kaSIaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGaam yzaKqbaoaaCaaaleqabaqcLbsacqGHsislcqaH4oqCcaWG4baaaaGc caGL7bGaayzFaaqcfa4aaWbaaSqabeaajugibiabeg7aHjabgkHiTi aaigdaaaaakiaawUfacaGLDbaaaSqaaKqzGeGaamyAaiabg2da9iaa igdaaSqaaKqzGeGaamOBaaGaey4dIunaaaa@95E8@

The log likelihood function is given by

logL( α,θ )=3nlogθ+nlogαnlog( θ 2 +θ+2 )+ i=1 n log ( 1+x+ x 2 )θ i=1 n x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciiBai aac+gacaGGNbGaamitaKqbaoaabmaabaqcLbsacqaHXoqycaGGSaGa eqiUdehajuaGcaGLOaGaayzkaaqcLbsacqGH9aqpcaaIZaGaamOBai GacYgacaGGVbGaai4zaiabeI7aXjabgUcaRiaad6gaciGGSbGaai4B aiaacEgacqaHXoqycqGHsislcaWGUbGaciiBaiaac+gacaGGNbqcfa 4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikda aaGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaqcLb sacqGHRaWkjuaGdaaeWbGcbaqcLbsaciGGSbGaai4BaiaacEgaaSqa aKqzGeGaamyAaiabg2da9iaaigdaaSqaaKqzGeGaamOBaaGaeyyeIu oajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIaamiEaiabgUcaRiaa dIhajuaGdaahaaWcbeqaaKqzGeGaaGOmaaaaaOGaayjkaiaawMcaaK qzGeGaeyOeI0IaeqiUdexcfa4aaabCaOqaaKqzGeGaamiEaaWcbaqc LbsacaWGPbGaeyypa0JaaGymaaWcbaqcLbsacaWGUbaacqGHris5aa aa@80B6@ +( α1 ) i=1 n log{ 1( 1+ θx(θx+θ+2) θ 2 +θ+2 ) e θx } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaey4kaS scfa4aaeWaaOqaaKqzGeGaeqySdeMaeyOeI0IaaGymaaGccaGLOaGa ayzkaaqcfa4aaabCaOqaaKqzGeGaciiBaiaac+gacaGGNbqcfa4aai WaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaabmaakeaajugibiaaigda cqGHRaWkjuaGdaWcaaGcbaqcLbsacqaH4oqCcaWG4bGaaiikaiabeI 7aXjaadIhacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaGaaiykaaGcbaqc LbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcq aH4oqCcqGHRaWkcaaIYaaaaaGccaGLOaGaayzkaaqcLbsacaWGLbqc fa4aaWbaaSqabeaajugibiabgkHiTiabeI7aXjaadIhaaaaakiaawU hacaGL9baaaSqaaKqzGeGaamyAaiabg2da9iaaigdaaSqaaKqzGeGa amOBaaGaeyyeIuoaaaa@6CCF@ (17)

For maximum likelihood estimates of parameters, we have to solve the following log-likelihood equations.

logL( α,θ ) α = n α + i=1 n log{ 1( 1+ θx(θx+θ+2) θ 2 +θ+2 ) e θx } =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aaS aaaOqaaKqzafGaeyOaIyRaciiBaiaac+gacaGGNbGaamitaKqbaoaa bmaabaGaeqySdeMaaiilaiabeI7aXbGaayjkaiaawMcaaaGcbaqcLb uacqGHciITcqaHXoqyaaGaeyypa0tcfa4aaSaaaOqaaKqzafGaamOB aaGcbaqcLbuacqaHXoqyaaGaey4kaScakeaajuaGdaaeWbGcbaqcLb uaciGGSbGaai4BaiaacEgajuaGdaGadaGcbaqcLbuacaaIXaGaeyOe I0scfa4aaeWaaOqaaKqzafGaaGymaiabgUcaRKqbaoaalaaakeaaju gqbiabeI7aXjaadIhacaGGOaGaeqiUdeNaamiEaiabgUcaRiabeI7a XjabgUcaRiaaikdacaGGPaaakeaajugqbiabeI7aXLqbaoaaCaaale qabaqcLbuacaaIYaaaaiabgUcaRiabeI7aXjabgUcaRiaaikdaaaaa kiaawIcacaGLPaaajugqbiaadwgajuaGdaahaaWcbeqaaKqzafGaey OeI0IaeqiUdeNaamiEaaaaaOGaay5Eaiaaw2haaaWcbaqcLbuacaWG PbGaeyypa0JaaGymaaWcbaqcLbuacaWGUbaacqGHris5aiabg2da9i aaicdaaaaa@7F30@

This gives

α ^ = n i=1 n log{ 1( 1+ θx(θx+θ+2) θ 2 +θ+2 ) e θx } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGUbaakeaajuaGdaae WbGcbaqcLbsaciGGSbGaai4BaiaacEgajuaGdaGadaGcbaqcLbsaca aIXaGaeyOeI0scfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRKqbaoaa laaakeaajugibiabeI7aXjaadIhacaGGOaGaeqiUdeNaamiEaiabgU caRiabeI7aXjabgUcaRiaaikdacaGGPaaakeaajugibiabeI7aXLqb aoaaCaaaleqabaqcLbsacaaIYaaaaiabgUcaRiabeI7aXjabgUcaRi aaikdaaaaakiaawIcacaGLPaaajugibiaadwgajuaGdaahaaWcbeqa aKqzGeGaeyOeI0IaeqiUdeNaamiEaaaaaOGaay5Eaiaaw2haaaWcba qcLbsacaWGPbGaeyypa0JaaGymaaWcbaqcLbsacaWGUbaacqGHris5 aaaaaaa@6AD5@

logL( α,θ ) θ = 3n θ n( 2θ+1 ) θ 2 +θ+2 i=1 n x +( α1 )ψ[ 1( 1+ θx(θx+θ+2) θ 2 +θ+2 ) e θx ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aaS aaaOqaaKqzGeGaeyOaIyRaciiBaiaac+gacaGGNbGaamitaKqbaoaa bmaabaqcLbsacqaHXoqycaGGSaGaeqiUdehajuaGcaGLOaGaayzkaa aakeaajugibiabgkGi2kabeI7aXbaacqGH9aqpjuaGdaWcaaGcbaqc LbsacaaIZaGaamOBaaGcbaqcLbsacqaH4oqCaaGaeyOeI0scfa4aaS aaaOqaaKqzGeGaamOBaKqbaoaabmaakeaajugibiaaikdacqaH4oqC cqGHRaWkcaaIXaaakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbao aaCaaaleqabaqcLbsacaaIYaaaaiabgUcaRiabeI7aXjabgUcaRiaa ikdaaaGaeyOeI0scfa4aaabCaOqaaKqzGeGaamiEaaWcbaqcLbsaca WGPbGaeyypa0JaaGymaaWcbaqcLbsacaWGUbaacqGHris5aaGcbaqc LbsacqGHRaWkjuaGdaqadaGcbaqcLbsacqaHXoqycqGHsislcaaIXa aakiaawIcacaGLPaaajugibiabeI8a5Lqbaoaadmaakeaajugibiaa igdacqGHsisljuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSscfa4aaS aaaOqaaKqzGeGaeqiUdeNaamiEaiaacIcacqaH4oqCcaWG4bGaey4k aSIaeqiUdeNaey4kaSIaaGOmaiaacMcaaOqaaKqzGeGaeqiUdexcfa 4aaWbaaSqabeaajugibiaaikdaaaGaey4kaSIaeqiUdeNaey4kaSIa aGOmaaaaaOGaayjkaiaawMcaaKqzGeGaamyzaKqbaoaaCaaaleqaba qcLbsacqGHsislcqaH4oqCcaWG4baaaaGccaGLBbGaayzxaaqcLbsa cqGH9aqpcaaIWaaaaaa@9A30@

where ψ( . ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiYdK xcfa4aaeWaaOqaaKqzGeGaaiOlaaGccaGLOaGaayzkaaaaaa@3BB4@ is a digamma function.

Hence, it is very difficult to estimate the value of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@ because the above-mentioned equation is too complicated. Therefore, we use sophisticated software like R for estimating the required parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@ .

Maximum product spacing estimation of parameters

The maximum product spacing estimates (MPSE) ( θ ^ , α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiqbeI7aXzaajaGaaiilaiqbeg7aHzaajaaakiaawIcacaGL Paaaaaa@3CCA@ of parameters ( θ,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabeI7aXjaacYcacqaHXoqyaOGaayjkaiaawMcaaaaa@3CAA@ of ESD can be obtained by numerically by maximizing the following function with respect to θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@ and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ .

MPSE= 1 n+1 i=1 n+1 log[ F( x i ,θ,α )F( x i1 ,θ,α ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytai aadcfacaWGtbGaamyraiabg2da9Kqbaoaalaaakeaajugibiaaigda aOqaaKqzGeGaamOBaiabgUcaRiaaigdaaaqcfa4aaabCaOqaaKqzGe GaciiBaiaac+gacaGGNbqcfa4aamWaaOqaaKqzGeGaamOraKqbaoaa bmaakeaajugibiaadIhalmaaBaaabaqcLbmacaWGPbaaleqaaKqzGe GaaiilaiabeI7aXjaacYcacqaHXoqyaOGaayjkaiaawMcaaKqzGeGa eyOeI0IaamOraKqbaoaabmaakeaajugibiaadIhalmaaBaaabaqcLb macaWGPbGaeyOeI0IaaGymaaWcbeaajugibiaacYcacqaH4oqCcaGG SaGaeqySdegakiaawIcacaGLPaaaaiaawUfacaGLDbaaaSqaaKqzad GaamyAaiabg2da9iaaigdaaSqaaKqzadGaamOBaiabgUcaRiaaigda aKqzGeGaeyyeIuoaaaa@6CB4@

Simulation study of ESD

In this section, we carried out simulation study to examine the performance of maximum likelihood estimators of the ESD. We examined the mean estimates, biases (B), mean square errors (MSEs) and variances of the maximum likelihood estimates (MLEs). The mean, bias, MSE and variance are computed using the formulae

Mean= 1 n i=1 n H ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytai aadwgacaWGHbGaamOBaiabg2da9Kqbaoaalaaakeaajugibiaaigda aOqaaKqzGeGaamOBaaaajuaGdaaeWbGcbaqcLbsaceWGibGbaKaaju aGdaWgaaWcbaqcLbmacaWGPbaaleqaaaqaaKqzadGaamyAaiabg2da 9iaaigdaaSqaaKqzadGaamOBaaqcLbsacqGHris5aaaa@4C3A@ , B= 1 n i=1 n ( H ^ i H ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqai abg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamOBaaaa juaGdaaeWbGcbaqcfa4aaeWaaOqaaKqzGeGabmisayaajaWcdaWgaa qaaKqzadGaamyAaaWcbeaajugibiabgkHiTiaadIeaaOGaayjkaiaa wMcaaaWcbaqcLbmacaWGPbGaeyypa0JaaGymaaWcbaqcLbmacaWGUb aajugibiabggHiLdaaaa@4D5D@ , MSE= 1 n i=1 n ( H ^ i H ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytai aadofacaWGfbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqc LbsacaWGUbaaaKqbaoaaqahakeaajuaGdaqadaGcbaqcLbsaceWGib GbaKaalmaaBaaabaqcLbmacaWGPbaaleqaaKqzGeGaeyOeI0Iaamis aaGccaGLOaGaayzkaaaaleaajugWaiaadMgacqGH9aqpcaaIXaaale aajugWaiaad6gaaKqzGeGaeyyeIuoalmaaCaaabeqaaKqzadGaaGOm aaaaaaa@5121@ , Variance=MSE B 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOvai aadggacaWGYbGaamyAaiaadggacaWGUbGaam4yaiaadwgacaaMc8Ua eyypa0JaamytaiaadofacaWGfbGaeyOeI0IaamOqaSWaaWbaaeqaba qcLbmacaaIYaaaaaaa@469B@

where H=θ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisai abg2da9iabeI7aXjaacYcacqaHXoqyaaa@3C52@ and H ^ i = θ ^ i , α ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmisay aajaWcdaWgaaqaaKqzadGaamyAaaWcbeaajugibiabg2da9iqbeI7a XzaajaWcdaWgaaqaaKqzadGaamyAaaWcbeaajugibiaacYcacuaHXo qygaqcaSWaaSbaaeaajugWaiaadMgaaSqabaaaaa@4499@

the simulation results for different parameter values of ESD are presented in tables 1 & 2 respectively. The steps for simulation study are as follows:

  1. Data is generated using the acceptance-rejection method of simulation. The acceptance-rejection method is a commonly used approach in simulation studies to generate random samples from a target distribution when inverse transform method of simulation is not feasible or efficient. Acceptance rejection method for generating random samples from the ESD consists of following steps.
    1. Generate Y distributed as exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabeI7aXbGccaGLOaGaayzkaaaaaa@3A5B@
    2. Generate U distributed as Uniform ( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaaicdacaGGSaGaaGymaaGccaGLOaGaayzkaaaaaa@3ACA@
    3. If U f(y) Mg(y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyvai abgsMiJMqbaoaalaaakeaajugibiaadAgacaGGOaGaamyEaiaacMca aOqaaKqzGeGaamytaiaaykW7caWGNbGaaiikaiaadMhacaGGPaaaaa aa@43BB@ , then set X=Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwai abg2da9iaadMfaaaa@393B@ (“accept the sample”); otherwise (“reject the sample”) and if reject then repeat the process: step (i-iii) until getting the required samples.
    4. Where M is a constant.

  2. The sample sizes are taken as n=50,100,150,200,250,300 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOBai abg2da9iaaiwdacaaIWaGaaiilaiaaigdacaaIWaGaaGimaiaacYca caaIXaGaaGynaiaaicdacaGGSaGaaGOmaiaaicdacaaIWaGaaiilai aaikdacaaI1aGaaGimaiaacYcacaaIZaGaaGimaiaaicdaaaa@4855@
  3. The parameter values are set as values θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde Naeyypa0JaaGymaaaa@39F1@ , α=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaaG4maaaa@39DC@ and θ=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aaaaa@3B62@ , α=6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaaGOnaaaa@39DF@
  4. Each sample size is replicated 10000 times

Sample size

Parameters

Mean

Bias

MSE

Variance

50

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.01898

0.01898

0.00235

0.00199

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

3.06882

0.06882

0.03697

0.03697

100

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.01400

0.01400

0.00261

0.00241

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

3.06173

0.06173

0.03074

0.02693

150

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.01326

0.01326

0.00221

0.00204

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

3.05937

0.05937

0.02867

0.02514

200

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.01250

0.01250

0.00213

0.00198

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

3.05253

0.05253

0.02620

0.02344

250

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.01103

0.01103

0.00202

0.00190

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

3.04911

0.04911

0.02441

0.02199

300

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.01089

0.01089

0.00190

0.00178

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

3.04626

0.04626

0.02347

0.02133

Table 1 The mean values, biases, MSEs and variances of ESD for parameter θ = 1, α = 3

Sample size

Parameters

Mean

Bias

MSE

Variance

50

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.50881

0.00881

0.00134

0.00127

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

6.12457

0.12457

0.05017

0.03465

100

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.50346

0.00346

0.00114

0.00113

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

6.11594

0.11594

0.05175

0.05089

150

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.50133

0.00133

0.00108

0.00107

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

6.11273

0.11273

0.05059

0.03788

200

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.50044

0.00044

0.00094

0.00094

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

6.09288

0.09288

0.04559

0.03696

250

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.49980

-0.00019

0.00089

0.00089

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

6.09316

0.09316

0.04067

0.03199

300

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaaaaa@3841@

1.49997

-0.00003

0.00080

0.00080

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaaaaa@382A@

6.09203

0.09203

0.03715

0.02868

Table 2 The mean values, biases, MSEs and variances of ESD for parameter θ = 1.5, α = 6

The results obtained in Tables 1 & 2 show that as the sample size increases, biases, MSEs and variances of the MLEs of the parameters become smaller respectively. This result is in line with the first-order asymptotic theory of maximum likelihood estimators.

Variance-covariance matrix for parameter θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde Naeyypa0JaaGymaaaa@39F1@ , α=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaaG4maaaa@39DC@ and θ=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aaaaa@3B62@ α=6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaaGOnaaaa@39DF@ ,

θ ^ α ^ θ ^ α ^ ( 0.00179 0.00027 0.00027 0.02140 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlqbeI7aXzaajaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7cuaHXoqygaqcaaGcbaqcLbsafaqabeGa baaakeaajugibiqbeI7aXzaajaaakeaajugibiqbeg7aHzaajaaaaK qbaoaabmaakeaajugibuaabeqaciaaaOqaaKqzGeaeaaaaaaaaa8qa caaIWaGaaiOlaiaaicdacaaIWaGaaGymaiaaiEdacaaI5aaak8aaba qcLbsapeGaaGimaiaac6cacaaIWaGaaGimaiaaicdacaaIYaGaaG4n aaGcpaqaaKqzGeWdbiaaicdacaGGUaGaaGimaiaaicdacaaIWaGaaG OmaiaaiEdaaOWdaeaajugib8qacaaIWaGaaiOlaiaaicdacaaIYaGa aGymaiaaisdacaaIWaaaaaGcpaGaayjkaiaawMcaaaaaaa@8DD5@ θ ^ α ^ θ ^ α ^ ( 0.00081 0.00016 0.00016 0.02877 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlqbeI7aXzaajaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7cuaHXoqygaqcaaGcbaqcLbsafaqabeGa baaakeaajugibiqbeI7aXzaajaaakeaajugibiqbeg7aHzaajaaaaK qbaoaabmaakeaajugibuaabeqaciaaaOqaaKqzGeaeaaaaaaaaa8qa caaIWaGaaiOlaiaaicdacaaIWaGaaGimaiaaiIdacaaIXaaak8aaba qcLbsapeGaaGimaiaac6cacaaIWaGaaGimaiaaicdacaaIXaGaaGOn aaGcpaqaaKqzGeWdbiaaicdacaGGUaGaaGimaiaaicdacaaIWaGaaG ymaiaaiAdaaOWdaeaajugibuaabeaabeaaaOqaaKqzGeWdbiaaicda caGGUaGaaGimaiaaikdacaaI4aGaaG4naiaaiEdaaaaaaaGcpaGaay jkaiaawMcaaaaaaa@8E7D@

Applications

The applications and the goodness of fit of the ESD have been discussed in this section with two examples of observed real datasets. The following datasets have been considered.

Dataset 1: The first dataset represents the breaking stress of carbon fibers of 50 mm length (GPa) reported by Nicolas and Padgett21, the dataset is presented below

0.39,0.85,1.08,1.25,1.47,1.57,1.61,1.61,1.69,1.80,1.84,1.87,1.89,2.03,2.03,2.05,2.12,2.35,
2.41,2.43,2.48,2.50,2.53,2.55,2.55,2.56,2.59,2.67,2.73,2.74,2.79,2.81,2.82,2.85,2.87,2.88,
2.93,2.95,2.96,2.97,3.09,3.11,3.11,3.15,3.15,3.19,3.22,3.22,3.27,3.28,3.31,3.31,3.33,3.39,3.39,
3.56,3.60,3.65,3.68,3.70,3.75,4.20,4.38,4.42,4.70,4.90.

Dataset 2: The following data represents the symmetric behavior of the tensile strength about 100 observations of carbon fibers, discussed by Nicolas and Padgett21the observations are:

3.7, 3.11, 4.42, 3.28, 3.75, 2.96, 3.39, 3.31, 3.15, 2.81, 1.41, 2.76, 3.19, 1.59, 2.17, 3.51, 1.84, 1.61, 1.57, 1.89, 2.74, 3.27, 2.41, 3.09, 2.43, 2.53, 2.81, 3.31, 2.35, 2.77, 2.68, 4.91, 1.57, 2.00, 1.17, 2.17, 0.39, 2.79, 1.08, 2.88, 2.73, 2.87, 3.19, 1.87, 2.95, 2.67, 4.20, 2.85, 2.55, 2.17, 2.97, 3.68, 0.81, 1.22, 5.08, 1.69, 3.68, 4.70, 2.03, 2.82, 2.50, 1.47, 3.22, 3.15, 2.97, 1.61, 2.05, 3.60, 3.11, 1.69, 4.90, 3.39, 3.22, 2.55, 3.56, 2.38, 1.92, 0.98, 1.59, 1.73, 1.71, 1.18, 4.38, 0.85, 1.80, 2.12, 3.65.

The figure 4 & 5 represents the total time on test (TTT) plot for the both observed two samples and simulated samples of ESD respectively. The figure 4 & 6 indicate that both the observed samples have decreasing failure rate.

Figure 4 TTT- plot of the observed sample-1and simulated samples of ESD respectively.

Figure 5 TTT- plot of the observed sample-2 and simulated samples of ESD respectively.

Figure 6 Fitted plot of distributions of the dataset-1 and dataset-2.

The pdf of Exponentiated Exponential distribution (EED) and exponentiated Aradhana distribution (EAD) are given by

f( x;θ,α )=αθ e θx ( 1 e θx ) α1 ;x>0,α>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaabmaakeaajugibiaadIhacaGG7aGaeqiUdeNaaiilaiabeg7a HbGccaGLOaGaayzkaaqcLbsacqGH9aqpcqaHXoqycqaH4oqCcaWGLb qcfa4aaWbaaSqabeaajugibiabgkHiTiabeI7aXjaadIhaaaqcfa4a aeWaaOqaaKqzGeGaaGymaiabgkHiTiaadwgajuaGdaahaaWcbeqaaK qzGeGaeyOeI0IaeqiUdeNaamiEaaaaaOGaayjkaiaawMcaaKqbaoaa CaaaleqabaqcLbsacqaHXoqycqGHsislcaaIXaaaaiaacUdacaWG4b GaeyOpa4JaaGimaiaacYcacqaHXoqycqGH+aGpcaaIWaGaaiilaiab eI7aXjabg6da+iaaicdaaaa@651C@

f( x;θ,α )= α θ 3 ( 1+x ) 2 e θx θ 2 +2θ+2 ( 1( 1+ θx( θx+2θ+2 ) θ 2 +2θ+2 ) e θx ) α1 ;x>0,α>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGMbqcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCcaGGSaGa eqySdegakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaakeaaju gibiabeg7aHjabeI7aXLqbaoaaCaaaleqabaqcLbsacaaIZaaaaKqb aoaabmaakeaajugibiaaigdacqGHRaWkcaWG4baakiaawIcacaGLPa aajuaGdaahaaWcbeqaaKqzGeGaaGOmaaaacaWGLbqcfa4aaWbaaSqa beaajugibiabgkHiTiabeI7aXjaadIhaaaaakeaajugibiabeI7aXL qbaoaaCaaaleqabaqcLbsacaaIYaaaaiabgUcaRiaaikdacqaH4oqC cqGHRaWkcaaIYaaaaKqbaoaabmaakeaajugibiaaigdacqGHsislju aGdaqadaGcbaqcLbsacaaIXaGaey4kaSscfa4aaSaaaOqaaKqzGeGa eqiUdeNaamiEaKqbaoaabmaakeaajugibiabeI7aXjaadIhacqGHRa WkcaaIYaGaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaabaqc LbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkca aIYaGaeqiUdeNaey4kaSIaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGa amyzaKqbaoaaCaaaleqabaqcLbsacqGHsislcqaH4oqCcaWG4baaaa GccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugibiabeg7aHjabgkHi TiaaigdaaaaakeaajugibiaacUdacaWG4bGaeyOpa4JaaGimaiaacY cacqaHXoqycqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6da+iaaicda aaaa@9607@

In order to compare lifetime distributions, values of 2logL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaGOmaiGacYgacaGGVbGaai4zaiaadYeaaaa@3BC4@ , Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion (CAIC), Hannan-Quinn Information Criterion (HQIC), Kolmogorov-Smirnov Statistics (K-S) and the corresponding probability value (p-value) for the above data set has been computed. The formulae for computing AIC, BIC, CAIC, HQIC and K-S are as follows:

AIC=2logL+2p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGVbGaai4z aiaadYeacqGHRaWkcaaIYaGaamiCaaaa@41B9@ , BIC=2logLplog( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGVbGaai4z aiaadYeacqGHsislcaWGWbGaciiBaiaac+gacaGGNbqcfa4aaeWaaO qaaKqzGeGaamOBaaGccaGLOaGaayzkaaaaaa@4786@ , CAIC=2logL+ 2pn np1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4qai aadgeacaWGjbGaam4qaiabg2da9iabgkHiTiaaikdaciGGSbGaai4B aiaacEgacaWGmbGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGOmaiaadc hacaWGUbaakeaajugibiaad6gacqGHsislcaWGWbGaeyOeI0IaaGym aaaaaaa@49C1@ , HQIC=2logL+2plog[ log( n ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisai aadgfacaWGjbGaam4qaiabg2da9iabgkHiTiaaikdaciGGSbGaai4B aiaacEgacaWGmbGaey4kaSIaaGOmaiaadchaciGGSbGaai4BaiaacE gajuaGdaWadaGcbaqcLbsaciGGSbGaai4BaiaacEgajuaGdaqadaGc baqcLbsacaWGUbaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@4EFC@ , K-S= Sup x | F m ( x ) F o ( x )| MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4sai aab2cacaWGtbGaeyypa0tcfa4aaCbeaOqaaKqzGeGaam4uaiaadwha caWGWbaaleaajugibiaadIhaaSqabaqcLbsacaGG8bGaamOraSWaaS baaeaajugWaiaad2gaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaaGc caGLOaGaayzkaaqcLbsacqGHsislcaWGgbWcdaWgaaqaaKqzadGaam 4BaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaa jugibiaacYhaaaa@51E4@

where, p=number of parameters, n=sample size, F m ( x )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraS WaaSbaaeaajugWaiaad2gaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaqcLbsacqGH9aqpaaa@3EE8@ empirical cdf of considered distribution and F o ( x )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraS WaaSbaaeaajugWaiaad+gaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaqcLbsacqGH9aqpaaa@3EEA@ cdf of considered distribution.

The distribution corresponding to the lower values of 2logL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaGOmaiGacYgacaGGVbGaai4zaiaadYeaaaa@3BC4@ , AIC, BIC, CAIC, HQIC and K-S Statistics and higher values of p-value is the best fit distribution. These statistical values for the two datasets have been computed and presented in tables 4 & 6 respectively. The estimation of the parameters using the method of MLE and MPSE for the two datasets is presented in the table 3 & 5 respectively.

Distributions

MLE

 

MPSE

θ ^  SE( θ ^ )   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaiaabccacaqGtbGaaeyraiaabIcacuaH4oqCgaqcaiaabMca caqGGaGaaeiiaaaa@3EEF@   α ^  SE( α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaeiiai qbeg7aHzaajaGaaeiiaiaabofacaqGfbGaaeikaiqbeg7aHzaajaGa aeykaaaa@3E14@ θ ^  SE( θ ^ )   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaiaabccacaqGtbGaaeyraiaabIcacuaH4oqCgaqcaiaabMca caqGGaGaaeiiaaaa@3EEF@

θ ^  SE( α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaacaqGGaGaae4uaiaabweacaqGOaGafqySdeMbaKaacaqGPaaa aa@3D88@

Sujatha

0.8534

0.8485 (0.0602)

 

(0.0607)

     

EED

1.0075

9.1992

0.8068

7.5849 (2.8926)

 

(0.1002)

(2.1491)

(0.0547)

 

EAD

1.4808

5.5104

1.4839 (0.1169)

5.2793

 

(0.1180)

(1.3217)

 

-1.2459

ESD

1.5248

5.9074

1.5276 (0.1177)

5.6613 (1.3664)

 

(0.1185)

(1.4465)

 

 

Table 3 MLE’s and MPSE’s with standard errors of the considered distributions for the dataset-1

Distributions             2logL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaGOmaiGacYgacaGGVbGaai4zaiaadYeaaaa@3BC5@

AIC

BIC

CAIC

HQIC

K-S 

P-value

Sujatha

229.59

231.59

233.78

231.65

232.45

0.2261

0.0052

EED

190.74

194.74

199.11

194.93

196.47

0.1589

0.1105

EAD

185.15

189.15

193.53

189.34

190.88

0.1599

0.1049

ESD

184.27

188.27

192.65

188.46

190.00

0.1404

0.4108

Table 4 Goodness of fit measures for the dataset-1

Distributions

MLE

 

                    MPSE

 

 

θ ^  SE( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaacaqGGaGaae4uaiaabweacaqGOaGafqiUdeNbaKaacaqGPaaa aa@3D9F@

α ^  SE( α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaacaqGGaGaae4uaiaabweacaqGOaGafqySdeMbaKaacaqGPaaa aa@3D71@

θ ^  SE( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaacaqGGaGaae4uaiaabweacaqGOaGafqiUdeNbaKaacaqGPaaa aa@3D9F@ /p>

α ^  SE( α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaacaqGGaGaae4uaiaabweacaqGOaGafqySdeMbaKaacaqGPaaa aa@3D71@

Sujatha

0.8863

0.8819

 

(0.0550)

 

(0.0546)

 

EED

1.0045

7.7135

0.8624

7.2864 (2.0352)

 

(0.0923)

(1.5724)

(0.0525)

 

EAD

1.4695

4.5305

1.5399 (0.1080)

4.9071 (0.9966)

 

(0.1084)

(0.9438)

   

ESD

1.5141

4.8594

1.5847 (0.1087)

5.2822 (1.0987)

 

(0.1088)

(1.0342)

 

 

Table 5 MLE’s and MPSE’s with standard errors of the considered distributions for the dataset-2

Distributions                    2logL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOeI0 IaaGOmaiGacYgacaGGVbGaai4zaiaadYeaaaa@3BC5@

AIC

BIC

CAIC

HQIC

K-S 

P-value

Sujatha

299.11

301.11

303.29

301.17

301.97

0.2115

0.0026

EED

254.81

258.81

263.18

259

260.54

0.0922

0.5653

EAD

250.36

254.36

258.74

254.55

256.09

0.1129

0.3084

ESD

249.70

253.70

258.07

253.89

255.43

0.0801

0.8731

Table 6 Goodness of fit measures for the dataset-2

From the table 4 &6 we observed that the ESD have the least -2logL, AIC, BIC, CAIC, HQIC and K-S values as compared to EED, EAD and Sujatha distribution. Hence, we may conclude that ESD provides a better fit than EED, EAD and Sujatha distribution. From the fitted plot and the P-P plot of the considered distribution presented in the figure 6, 7 & 8 for the both datasets also exhibit that ESD provides a better fit as compared to the considered distributions.

Figure 7 P-P plots of the theoretical and sample quantiles of the considered distributions of the dataset-1.

Figure 8 P-P plots of the theoretical and sample quantiles of the considered distributions of the dataset-2.

Conclusions

In this paper an extended Sujatha distribution (ESD) has been proposed. Its statistical properties including survival function, hazard function, harmonic mean, moment generating function, order statistics and Renyi entropy have been discussed. Moments of the proposed distribution has been obtained. The parameters of this distribution have been estimated using maximum likelihood estimation method and maximum product spacing method. The simulation study has been presented to know the performance of maximum likelihood estimates as the sample size increases. Finally, two examples of real lifetime datasets have been considered for applications and compared with the EED, EAD and Sujatha distribution. It has been found that ESD provides the best fit than the EED, EAD and Sujatha distribution.

Funding information

No funding was received from any financial organization to conduct this research.

Acknowledgments

Authors are grateful to the editor-in-chief and the anonymous reviewer for their comments which improved the quality of the paper.

Declaration of competing interest

The authors declare that they have no any known financial or non-financial competing interests in any material discussed in this paper.

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