# Inference on Exponentiated Power Lindley Distribution Based on Order Statistics with Application

Authors: Mansour Shrahili; Naif Alotaibi; Devendra Kumar; A. R. Shafay
Journal: Complexity
 (2020)

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## Abstract

Exponentiated power Lindley distribution is proposed as a generalization of some widely well-known distributions such as Lindley, power Lindley, and generalized Lindley distributions. In this paper, the exact explicit expressions for moments of order statistics from the exponentiated power Lindley distribution are derived. By using these relations, the best linear unbiased estimates of the location and scale parameters, based on type-II right-censored sample, are obtained. Next, the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics are calculated and then used to derive the approximate confidence interval for the location and scale parameters using the Edgeworth approximation. Finally, some numerical illustrations and two real data applications are presented.


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## Body

## 1. Introduction

Ashour and Eltehiwy [1] introduced the exponentiated power Lindley (EPL) distribution as a generalization of two-parameter power Lindley distribution and they studied some mathematical properties of the EPL distribution. They also showed that this distribution provides more flexibility to analyze a complex real data set. Due to its practicality, the EPL distribution can be used for many applications, including accelerated life testing, survival analysis, reliability, biology, and others. The EPL distribution has attractive feature that its hazard rate function could be decreasing, increasing, and decreasing-increasing-decreasing but not constant, depending on the shape parameter. The EPL distribution has a unimodal and right skewed probability density function (PDF) for δ>1. Many authors have developed generalization of Lindley distribution. Prominent among these generalizations which we are aware of are generalized Lindley distribution by Nadarajah et al. [2]; power Lindley distribution by Ghitany et al. [3]; generalized inverse Lindley distribution by Sharma et al. [4]; pseudo-Lindley distribution by Nedjar and Zeghdoudi [5]; size-biased gamma Lindley distribution by Beghriche and Zeghdoud [6]. Ashour and Eltehiwy [1] observed that in numerous situations, the EPL distribution provided a better fit than Lindley, power Lindley, generalized Lindley, exponentiated exponential, modified Weibull, and Weibull distributions.Over the past six decades or so, several authors have shown keen interest in order statistics. These order statistics and their moments are of great significance in many real life applications involving data relating to flood, drought, reliability, engineering, and life testing situation. In recent times, several papers and books have been published on order statistics and their distributional properties. Among them are David and Nagaraja [7]; Arnold et al. [8]; Balakrishnan and Cohen [9]; Balakrishnan and Ahsanullah [10]; Balakrishnan and Sultan [11]; Malik et al. [12]; Mahmoud et al. [13]; Genc [14]; MirMostafaee [15]; Balakrishnan et al. [16]; Kumar and Dey [17]; Sultan et al. [18]; Sultan and AL-Thubyani [19]; Kumar et al. [20]; and so on. Due to the wide applications of order statistics, several authors have carried out extensive studies on these kinds of ordered data. Numerous papers dealing with moments and estimation of parameters of different lifetime distributions based on order statistics can be found in literature. For example, Balakrishnan and Cohen [9] discussed the moments and estimation of parameters. Sultan et al. [18] obtained the moments of power function distribution. Sultan and AL-Thubyani [19] developed the higher-order moments and inferential procedure for estimating parameters of Lindley distribution. Ahsanullah and Alzaatreh [21] studied the moments and inferential procedure for log-logistic distribution. Recently, Kumar and Goyal [22, 23] and Kumar et al. [24, 25] developed inferential procedure for power Lindley and generalized Lindley, modified power function, and generalized inverse Lindley distribution.LetY1,Y2,…,Ym be an independent random variable from the EPL distribution with cumulative distribution function (CDF) and PDF is given, respectively, by(1)Fy=1−1+υ+υyγ1+υexp−υyγδ,y>0,δ>0,υ>0,γ>0,(2)fy=δυ2γyγ−11+υ1+yγexp−υyγ1−1+υ+υyγ1+υexp−υyγδ−1,y>0.The hazard function of the EPL distribution is given by(3)hy=δυ2γyγ−11+υ1+yγVyδ−11−Vδy−1exp−υyγ,y>0,δ,υ,γ>0,where(4)Vy=1−1+υ+υy1+υexp−υyγ.Some widely well-known distributions can be obtained as special cases from the EPL distribution. By settingγ=1 and δ=1 in both (1) and (2), the CDF and PDF of the generalized Lindley distribution and power Lindley distribution are obtained, respectively, while setting δ=γ=1 gives the CDF and PDF of Lindley distributions.To the best of our knowledge, there are no reports on moments and estimation for the EPL distribution using order statistics. In this paper, first is to obtain the exact explicit expressions for moments of order statistics from the EPL distribution, and second is to derive the best linear unbiased estimates (BLUEs) of the location and scale parameters of the EPL distribution based on order statistics.The rest of this paper is organized as follows: in Section2, the moments of EPL distribution are derived based on order statistics. The BLUEs of the location and scale parameters are provided in Section 3. Next, these BLUEs are used in Section 4 to obtain the coefficient of skewness and kurtosis of some pivotal quantities. Also, we obtain Edgeworth approximations for the distributions of these pivotal quantities. Simulation study and real data analysis are presented in Sections 5 and 6. Finally, the paper ends with a conclusion in Section 7.


## 2. Moments of Order Statistics

In this section, the moments of order statistics are established for a given random sampleY1,Y2,…,Ym from the EPL distribution.


### 2.1. Single Moments

In this subsection, the explicit expressions for single moments of order statistics are derived. LetY1:m≤Y2:m≤⋯≤Ym:m be the order statistics for a given random sample Y1,Y2,…,Ym from the EPL distribution, and Yr:m, r=1,2,…,m, denotes the rth order statistics with PDF which is given by (see [7])(5)fYr:mx=Cr:mFr−1y1−Fym−rfy,−∞<y<∞,where F. and f. are given by (1) and (2), respectively, and(6)Cr:m=m!r−1!m−r!.Theorem 1.
For the EPL distribution given in (1), the p1th moment of the rth order statistics μr:mp1, p1=0,1,… and 1≤r≤m, is as follows:(7)μr:mp1=Cr:mδυ21+υ∑j1=0m−r∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×δr+j1−1j2j2j3j3+1j4Γp1+γj4+1/γυj2+1p1+γj4+1/γ.Proof.
From (5), we have(8)μr:mp1=EYr:mp1=Cr:m∫0∞yp1Fr−1y1−Fym−rfy=Cr:m∑j1=0m−r−1j1m−rj1∫0∞yp1Fyj1+r−1fydy.
By using (1) and (2), one can write(9)μr:mp1=Cr:mδγυ21+υ∑j1=0m−r−1j1m−rj1∫0∞yp1+γ−11+yγe−υyγ×1−1+υ+υyγ1+υe−υyγδr+j1−1dy=Cr:mδγυ21+υ∑j1=0m−r∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1δr+j1−1j2×j2j3j3+1j4∫0∞yp1+γj4+1−1e−υj2+1yγdy,hence the result.
Some special cases from equation (7):(1)
Ifδ=1, we get(10)μr:mp1=Cr:mυ21+υ∑j1=0m−r∑j2=0r+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×r+j1−1j2j2j3j3+1j4Γp1+γj4+1/γυj2+1p1+γj4+1/γ.(2)
Ifγ=1, we get(11)μr:mp1=Cr:mδυ21+υ∑j1=0m−r∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×δr+j1−1j2j2j3j3+1j4Γp1+j4+1υj2+1p1+j4+1,as obtained by Kumar and Goyal [23].(3)
Ifγ=1 and δ=1, we get(12)μr:mp1=Cr:mυ21+υ∑j1=0m−r∑j2=0r+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×r+j1−1j2j2j3j3+1j4Γp1+j4+1υj2+1p1+j4+1.(4)
Ifp1=1 and r=1, we get(13)μ1:m1=mδυ21+υ∑j1=0m−1∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−1j1×δj1+1−1j2j2j3j3+1j4Γ1+γj4+1/γυj2+11+γj4+1/γ.(5)
Ifp1=1 and r=m, we get(14)μm:m1=mδυ21+υ∑j2=0δm−1∑j3=0j2∑j4=0j3+1−1j2υj31+υj2×δm−1j2j2j3j3+1j4Γ1+γj4+1/γυj2+11+γj4+1/γ.(6)
Ifp1=1 and m=r=1, we get(15)μ1:11=δυ21+υ∑j2=0δ−1∑j3=0j2∑j4=0j3+1−1j2υj31+υj2δ−1j2j2j3j3+1j4Γγj4+1+1/γυj2+1γj4+1+1/γ=EY,as obtained by Ashour and Eltehiwy [1].


### 2.2. Double Moments

LetY1:m≤Y2:m≤⋯≤Yn:m be the order statistics from the EPL distribution. Then, the joint PDF of the rth and sth order statistics is given as(16)fYr:m,Ys:my,z=Cr,s:mFr−1yFz−Fys−1−r1−Fzm−sfyfz,0<y<z,where 1≤r<s≤m and Cr,s:m=m!/r−1!s−r−1!m−s!.Theorem 2.
For the EPL distribution given in (1), the double moment of rth and sth order statistics, μr,s:mp1,p2, p1,p2=1,2,…, is as follows:(17)μr,s:mp1,p2=Cr,s:mδ2υ41+υ∑j1=0s−r−1∑j2=0m−s∑j3=0δr+j1−1∑i4=0δs−r+j2−j1−1∑j5=0j3∑i6=0j4∑i7=0j5+1∑j8=0j6+1∑j9=0p1+γj8+1/γ−1×−1j1+j2+j3+j4υj5+j61+υj1+j3+j4s−r−1j1m−sj2δr+j1−1j3×δs−r+j2−j1−1j4j3j5j4j6j5+1j7j6+1j8×υj4+1j9−p1+γj8+1/γp1+γj8+1/γ−1!j9!Γp1+γj8+1/γ+1υj3+j4+2p1+γj8+1/γ+1.Proof.
Using (16), we have(18)μr,s:mp1,p2=EYr,s:mp1,p2=Cr,s:m∫0∞∫y∞yp1zp2Fyr−1Fz−Fys−r−11−Fzn−sfyfzdzdy.
By using the same argument as in Theorem1, we get the result given in (17).


### 2.3. Triple Moments

LetY1:m≤Y2:m≤⋯≤Yn:m denote the order statistics drawn from the EPL distribution. The joint PDF of the rth, sth, and tth order statistics is given by(19)fYr:m,Ys:m,Yt:my,z,u=Cr,s,t:mFr−1yFz−Fys−1−rFu−Fzt−1−s×1−Fum−tfyfzfu,0<y<z<u,where 1≤r<s<t≤m and(20)Cr,s,t:m=m!r−1!s−r−1!t−s−1!m−t!.Theorem 3.
For the EPL distribution given in (1), The triple moment of rth, sth, and tth order statistics μr,s,t:mp1,p2,p3, p1,p2,p3=1,2,…, is as follows:(21)μr,s,t:mp1,p2,p3=δ3Cr,s,t:m∑ℓ1=0s−1−r∑ℓ2=0t−1−s∑ℓ3=0m−ts−1−rℓ1t−1−sℓ2m−tℓ3−1ℓ1+ℓ2+ℓ3×∑j1=0δt−s−ℓ2+ℓ3−1∑j2=0j1∑j3=0j2+1∑j4=0p3/γ+j3∑j5=0δs−r−ℓ1+ℓ2−1∑j6=0j5∑j7=0j6+1∑j8=0p2/γ+j4+j7∑j9=0δr+ℓ1−1∑j10=0j9∑j11=0j10+1×δt−s−ℓ2+ℓ3−1j1j1j2j2+1j3δs−r−ℓ1+ℓ2−1j5j5j6j6+1j7×δr+ℓ1−1j9j9j10j10+1j11−1j1+j5+j9υj2−j3+j6−j7+j10−j11−p1+p2+p3/γ+3j4!j8!1+υj1+j5+j9+3j1+1l/β+j3−j4+1×p3/γ+j3!p2/γ+j4+j7!Γp1/γ+j8+j11+1j5+j1+2p2/γ+j4+j7−j8+1j9+j5+j1+3p1/γ+j8+j11+1.Proof.
From equation (19), we have(22)μr,s,t:mp1,p2,p3=EYr:mp1Ys:mp2Yt:mp3=Cr,s,t:m∫0∞∫y∞∫z∞yp1zp2up3Fr−1yFz−Fys−r−1×Fu−Fzt−s−11−Fum−tfyfzfududzdy.
By using the same argument as in Theorems1 and 2, we obtain the result given in (21).


### 2.4. Quadruple Moments

LetY1:m≤Y2:m≤⋯≤Yn:m denote the order statistics drawn from the EPL distribution. The joint PDF of the rth, sth, tth, and wth order statistics is given as(23)fYr:m,Ys:m,Yt:m,Yw:my,z,u,v=Cr,s,t,w:mFr−1yFz−Fys−1−rFu−Fzt−1−s×Fv−Fuw−1−t1−Fvm−wfyfzfufv,where 1≤r<s<t<u≤m and(24)Cr,s,t,w:m=m!r−1!s−r−1!t−s−1!w−t−1m−w!.Theorem 4.
For the EPL distribution given in (1), the quadruple moment of rth, sth, tth, and wth order statistics μr,s,t:mp1,p2,p3,p4 is as follows:(25)μr,s,t:mp1,p2,p3,p4=δ4Cr,s,t,u:m∑ℓ1=0s−1−r∑ℓ2=0t−1−s∑ℓ3=0u−1−t∑ℓ4=0m−us−1−rℓ1t−1−sℓ2u−1−tℓ3m−uℓ4×−1ℓ1+ℓ2+ℓ3+ℓ4∑j1=0δw−t−ℓ3+ℓ4−1∑j2=0j1∑j3=0j2+1∑j4=0p4/γ+j3∑j5=0δt−s−ℓ2+ℓ3−1∑j6=0j5∑j7=0j6+1∑j8=0p3/γ+j4+j7×∑j9=0δs−r−ℓ1+ℓ2−1∑j10=0j9∑j11=0j10+1∑i12=0p2/γ+j8+j11∑j13=0δr+ℓ1−1∑j14=0j13∑j15=0i14+1δw−t−ℓ3+ℓ4−1j1×j1j2j2+1j3δt−s−ℓ2+ℓ3−1j5j5j6j6+1j7δs−r−ℓ1+ℓ2−1j9×j9j10j10+1j11δr+ℓ1−1j13j13j14j14+1j15×−1j1+j5+j9+j13υj2−j3+j6−j7+j10−j11+j14−j15−p1+p2+p3+p4/γ+4j4!j8!j12!1+υj1+j5+j9+j13+4j1+1p4/γ+j3−j4+1×p4/γ+j3!p3/γ+j4+j7!p2/γ+j8+j11!j5+j1+2p4/γ+j4+j7−j8+1j9+j5+j1+3p2/γ+j8+j11−j12+1×Γp1/γ+j12+j15+1j13+j9+j5+j1+4p1/γ+j12+j15+1.Proof.
From equation (23), we have(26)μr,s,t:mp1,p2,p3,p4=EYr:mp1Ys:mp2Yt:mp3Yw:mp4=Cr,s,t,w:m∫0∞∫y∞∫z∞∫u∞yp1zp2up3vp4Fz−Fys−1−r×Fr−1yFu−Fzt−1−sFv−Fuw−1−t1−Fvm−wfyfvfufvdvdudzdy.
By using the same argument as in Theorem3, we get the result given in (25).


## 2.1. Single Moments

In this subsection, the explicit expressions for single moments of order statistics are derived. LetY1:m≤Y2:m≤⋯≤Ym:m be the order statistics for a given random sample Y1,Y2,…,Ym from the EPL distribution, and Yr:m, r=1,2,…,m, denotes the rth order statistics with PDF which is given by (see [7])(5)fYr:mx=Cr:mFr−1y1−Fym−rfy,−∞<y<∞,where F. and f. are given by (1) and (2), respectively, and(6)Cr:m=m!r−1!m−r!.Theorem 1.
For the EPL distribution given in (1), the p1th moment of the rth order statistics μr:mp1, p1=0,1,… and 1≤r≤m, is as follows:(7)μr:mp1=Cr:mδυ21+υ∑j1=0m−r∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×δr+j1−1j2j2j3j3+1j4Γp1+γj4+1/γυj2+1p1+γj4+1/γ.Proof.
From (5), we have(8)μr:mp1=EYr:mp1=Cr:m∫0∞yp1Fr−1y1−Fym−rfy=Cr:m∑j1=0m−r−1j1m−rj1∫0∞yp1Fyj1+r−1fydy.
By using (1) and (2), one can write(9)μr:mp1=Cr:mδγυ21+υ∑j1=0m−r−1j1m−rj1∫0∞yp1+γ−11+yγe−υyγ×1−1+υ+υyγ1+υe−υyγδr+j1−1dy=Cr:mδγυ21+υ∑j1=0m−r∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1δr+j1−1j2×j2j3j3+1j4∫0∞yp1+γj4+1−1e−υj2+1yγdy,hence the result.
Some special cases from equation (7):(1)
Ifδ=1, we get(10)μr:mp1=Cr:mυ21+υ∑j1=0m−r∑j2=0r+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×r+j1−1j2j2j3j3+1j4Γp1+γj4+1/γυj2+1p1+γj4+1/γ.(2)
Ifγ=1, we get(11)μr:mp1=Cr:mδυ21+υ∑j1=0m−r∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×δr+j1−1j2j2j3j3+1j4Γp1+j4+1υj2+1p1+j4+1,as obtained by Kumar and Goyal [23].(3)
Ifγ=1 and δ=1, we get(12)μr:mp1=Cr:mυ21+υ∑j1=0m−r∑j2=0r+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−rj1×r+j1−1j2j2j3j3+1j4Γp1+j4+1υj2+1p1+j4+1.(4)
Ifp1=1 and r=1, we get(13)μ1:m1=mδυ21+υ∑j1=0m−1∑j2=0δr+j1−1∑j3=0j2∑j4=0j3+1−1j1+j2υj31+υj2m−1j1×δj1+1−1j2j2j3j3+1j4Γ1+γj4+1/γυj2+11+γj4+1/γ.(5)
Ifp1=1 and r=m, we get(14)μm:m1=mδυ21+υ∑j2=0δm−1∑j3=0j2∑j4=0j3+1−1j2υj31+υj2×δm−1j2j2j3j3+1j4Γ1+γj4+1/γυj2+11+γj4+1/γ.(6)
Ifp1=1 and m=r=1, we get(15)μ1:11=δυ21+υ∑j2=0δ−1∑j3=0j2∑j4=0j3+1−1j2υj31+υj2δ−1j2j2j3j3+1j4Γγj4+1+1/γυj2+1γj4+1+1/γ=EY,as obtained by Ashour and Eltehiwy [1].


## 2.2. Double Moments

LetY1:m≤Y2:m≤⋯≤Yn:m be the order statistics from the EPL distribution. Then, the joint PDF of the rth and sth order statistics is given as(16)fYr:m,Ys:my,z=Cr,s:mFr−1yFz−Fys−1−r1−Fzm−sfyfz,0<y<z,where 1≤r<s≤m and Cr,s:m=m!/r−1!s−r−1!m−s!.Theorem 2.
For the EPL distribution given in (1), the double moment of rth and sth order statistics, μr,s:mp1,p2, p1,p2=1,2,…, is as follows:(17)μr,s:mp1,p2=Cr,s:mδ2υ41+υ∑j1=0s−r−1∑j2=0m−s∑j3=0δr+j1−1∑i4=0δs−r+j2−j1−1∑j5=0j3∑i6=0j4∑i7=0j5+1∑j8=0j6+1∑j9=0p1+γj8+1/γ−1×−1j1+j2+j3+j4υj5+j61+υj1+j3+j4s−r−1j1m−sj2δr+j1−1j3×δs−r+j2−j1−1j4j3j5j4j6j5+1j7j6+1j8×υj4+1j9−p1+γj8+1/γp1+γj8+1/γ−1!j9!Γp1+γj8+1/γ+1υj3+j4+2p1+γj8+1/γ+1.Proof.
Using (16), we have(18)μr,s:mp1,p2=EYr,s:mp1,p2=Cr,s:m∫0∞∫y∞yp1zp2Fyr−1Fz−Fys−r−11−Fzn−sfyfzdzdy.
By using the same argument as in Theorem1, we get the result given in (17).


## 2.3. Triple Moments

LetY1:m≤Y2:m≤⋯≤Yn:m denote the order statistics drawn from the EPL distribution. The joint PDF of the rth, sth, and tth order statistics is given by(19)fYr:m,Ys:m,Yt:my,z,u=Cr,s,t:mFr−1yFz−Fys−1−rFu−Fzt−1−s×1−Fum−tfyfzfu,0<y<z<u,where 1≤r<s<t≤m and(20)Cr,s,t:m=m!r−1!s−r−1!t−s−1!m−t!.Theorem 3.
For the EPL distribution given in (1), The triple moment of rth, sth, and tth order statistics μr,s,t:mp1,p2,p3, p1,p2,p3=1,2,…, is as follows:(21)μr,s,t:mp1,p2,p3=δ3Cr,s,t:m∑ℓ1=0s−1−r∑ℓ2=0t−1−s∑ℓ3=0m−ts−1−rℓ1t−1−sℓ2m−tℓ3−1ℓ1+ℓ2+ℓ3×∑j1=0δt−s−ℓ2+ℓ3−1∑j2=0j1∑j3=0j2+1∑j4=0p3/γ+j3∑j5=0δs−r−ℓ1+ℓ2−1∑j6=0j5∑j7=0j6+1∑j8=0p2/γ+j4+j7∑j9=0δr+ℓ1−1∑j10=0j9∑j11=0j10+1×δt−s−ℓ2+ℓ3−1j1j1j2j2+1j3δs−r−ℓ1+ℓ2−1j5j5j6j6+1j7×δr+ℓ1−1j9j9j10j10+1j11−1j1+j5+j9υj2−j3+j6−j7+j10−j11−p1+p2+p3/γ+3j4!j8!1+υj1+j5+j9+3j1+1l/β+j3−j4+1×p3/γ+j3!p2/γ+j4+j7!Γp1/γ+j8+j11+1j5+j1+2p2/γ+j4+j7−j8+1j9+j5+j1+3p1/γ+j8+j11+1.Proof.
From equation (19), we have(22)μr,s,t:mp1,p2,p3=EYr:mp1Ys:mp2Yt:mp3=Cr,s,t:m∫0∞∫y∞∫z∞yp1zp2up3Fr−1yFz−Fys−r−1×Fu−Fzt−s−11−Fum−tfyfzfududzdy.
By using the same argument as in Theorems1 and 2, we obtain the result given in (21).


## 2.4. Quadruple Moments

LetY1:m≤Y2:m≤⋯≤Yn:m denote the order statistics drawn from the EPL distribution. The joint PDF of the rth, sth, tth, and wth order statistics is given as(23)fYr:m,Ys:m,Yt:m,Yw:my,z,u,v=Cr,s,t,w:mFr−1yFz−Fys−1−rFu−Fzt−1−s×Fv−Fuw−1−t1−Fvm−wfyfzfufv,where 1≤r<s<t<u≤m and(24)Cr,s,t,w:m=m!r−1!s−r−1!t−s−1!w−t−1m−w!.Theorem 4.
For the EPL distribution given in (1), the quadruple moment of rth, sth, tth, and wth order statistics μr,s,t:mp1,p2,p3,p4 is as follows:(25)μr,s,t:mp1,p2,p3,p4=δ4Cr,s,t,u:m∑ℓ1=0s−1−r∑ℓ2=0t−1−s∑ℓ3=0u−1−t∑ℓ4=0m−us−1−rℓ1t−1−sℓ2u−1−tℓ3m−uℓ4×−1ℓ1+ℓ2+ℓ3+ℓ4∑j1=0δw−t−ℓ3+ℓ4−1∑j2=0j1∑j3=0j2+1∑j4=0p4/γ+j3∑j5=0δt−s−ℓ2+ℓ3−1∑j6=0j5∑j7=0j6+1∑j8=0p3/γ+j4+j7×∑j9=0δs−r−ℓ1+ℓ2−1∑j10=0j9∑j11=0j10+1∑i12=0p2/γ+j8+j11∑j13=0δr+ℓ1−1∑j14=0j13∑j15=0i14+1δw−t−ℓ3+ℓ4−1j1×j1j2j2+1j3δt−s−ℓ2+ℓ3−1j5j5j6j6+1j7δs−r−ℓ1+ℓ2−1j9×j9j10j10+1j11δr+ℓ1−1j13j13j14j14+1j15×−1j1+j5+j9+j13υj2−j3+j6−j7+j10−j11+j14−j15−p1+p2+p3+p4/γ+4j4!j8!j12!1+υj1+j5+j9+j13+4j1+1p4/γ+j3−j4+1×p4/γ+j3!p3/γ+j4+j7!p2/γ+j8+j11!j5+j1+2p4/γ+j4+j7−j8+1j9+j5+j1+3p2/γ+j8+j11−j12+1×Γp1/γ+j12+j15+1j13+j9+j5+j1+4p1/γ+j12+j15+1.Proof.
From equation (23), we have(26)μr,s,t:mp1,p2,p3,p4=EYr:mp1Ys:mp2Yt:mp3Yw:mp4=Cr,s,t,w:m∫0∞∫y∞∫z∞∫u∞yp1zp2up3vp4Fz−Fys−1−r×Fr−1yFu−Fzt−1−sFv−Fuw−1−t1−Fvm−wfyfvfufvdvdudzdy.
By using the same argument as in Theorem3, we get the result given in (25).


## 3. Estimation of Parameters

Here, we study parameter estimation for the EPL distribution based on order statistics.


### 3.1. BLUEs of Parameters

The PDF of the scale-parameter EPL distribution is(27)fy=δγυ2σ1+υyσγ−11+yσγ1−1+υ+υy/σγ1+υe−υy/σγδ−1e−υy/σγ,y>0,δ,γ,υ,σ>0,and the PDF of the location-scale parameter EPL distribution is(28)fy=δγυ2σ1+υy−μσγ−11+y−μσγe−υy−μ/σγ×1−1+υ+υy−μ/σγ1+υe−υy−μ/σγδ−1,y>0,δ,γ,υ,σ,μ>0.LetY1:m≤Y2:m≤⋯≤Yn−c:m, c=0,1,…,n−1 denote type-II right-censored sample from the location-scale parameter EPL distribution in equation (1). Let us denote Xr:m=Yr:m−μ/σ, EXr:m=μr;m1, 1≤r≤m−c, and CovXr:m,Xs:m=σr,s:m=μr,s:m1,1−μr:m1μs:m1, 1≤r<s≤m−c. Therefore,(29)Y=Y1:m,Y2:m,…,Ym−c:mT,μ=μ1:m,μ2:m,…,μm−c:mT,1=1,1,…,1T︸m−c,∑=σr,s,1≤r,s≤m−c,where μi:m=EYi:m, σii=VarYi:m and σij=CovYi:m,Yj:m, and i=1,2,…m−c. Then, the BLUEs of μ and σ can be computed as follows (see Arnold et al. [8]):(30)μ∗=∑r=1m−cprYr:m,σ∗=∑r=1m−cqrYr:m,where(31)pr=μT∑−1μ1T∑−1−μT∑−11μT∑−1μT∑−1μ1T∑−11−μT∑−112(32)qr=1T∑−11μT∑−1−1T∑−1μ1T∑−1μT∑−1μ1T∑−11−μT∑−112.The variances and covariance of these BLUEs can be computed as follows (see Arnold et al. [8]):(33)Varμ∗=σ2μT∑−1μμT∑−1μ1T∑−11−μT∑−112=σ2W1,Varσ∗=σ21T∑−11μT∑−1μ1T∑−11−μT∑−112=σ2W2,Covμ∗,σ∗=σ2−μT∑−11μT∑−1μ1T∑−11−μT∑−112=σ2W3.The values ofpr and qr are displayed in Tables 1 and 2 for different values of sample sizes n=7,10 and different censoring cases c=01n/2−1 and for some selected values for υ=1,2. The coefficient of the BLUEs pr and qr are given by (17) and (18), respectively, with conditions(34)∑i=1n−cpi=1,∑i=1n−cqi=0,which are used to check the computations accuracy.Table 1
Coefficients of the BLUEs of the location parameter.
υncpi, i=1,2,…,n−c1701.1367330.005506−0.023661−0.026954−0.024014−0.032131−0.03548211.161942−0.000791−0.040361−0.029379−0.039803−0.05160321.215986−0.044508−0.036811−0.061522−0.0731461001.092681−0.0146520.027777−0.023314−0.001847−0.019939−0.014359−0.014698−0.014324−0.01732511.105425−0.0051280.000896−0.0176790.001242−0.025909−0.023216−0.012611−0.02302221.129876−0.005171−0.017773−0.019798−0.008271−0.031966−0.019298−0.02760131.1367330.005506−0.023616−0.026954−0.024014−0.032131−0.03548241.161942−0.000796−0.040361−0.029379−0.039803−0.0516032701.1318010.004217−0.030454−0.006347−0.030267−0.030084−0.03886711.164716−0.015401−0.017184−0.042061−0.038315−0.05175521.204871−0.030066−0.038861−0.061012−0.0749331001.0780950.018066−0.0127110.010341−0.018799−0.013232−0.014746−0.014122−0.014181−0.01872211.0888180.021605−0.014117−0.007688−0.019094−0.010235−0.016854−0.018684−0.02369921.1071220.015291−0.019433−0.017916−0.008332−0.019141−0.029435−0.02815731.1318010.004217−0.030454−0.006347−0.030267−0.030084−0.03886741.164716−0.015401−0.017184−0.04206−0.038315−0.0517553701.1259910.003349−0.003324−0.026256−0.035498−0.02695−0.03731111.158534−0.006501−0.023706−0.036411−0.039839−0.05207821.203149−0.025636−0.048175−0.053915−0.0753871001.084513−0.0010820.012616−0.006894−0.0256820.004272−0.023087−0.010631−0.015161−0.01886511.0896080.0057370.014226−0.023515−0.010205−0.013899−0.022786−0.016716−0.02245121.113338−0.0069220.010884−0.026283−0.014407−0.026124−0.022595−0.02789231.1259910.003349−0.003324−0.026256−0.035498−0.026951−0.03731141.158534−0.006501−0.023706−0.036411−0.039839−0.052078Table 2
Coefficients of the BLUEs of the scale parameter.
υncqi, i=1,2,…,n−c170−0.6071910.0492650.1073570.0985630.1060760.1151310.1307981−0.6190960.0651740.1482120.1166840.1332730.1557532−0.6507290.1423760.1601710.1540250.194157100−0.6153920.0815140.0287930.0381890.0747110.0695230.0770510.0870840.0712460.0872831−0.6231590.0890410.0272650.0798450.0644780.0872260.0947280.0824620.0981152−0.6108790.0671920.0426660.1080530.0812350.1026640.0977070.1113633−0.6071910.0492650.1073570.0985630.1060760.1151310.1307984−0.6190960.0651740.1482120.1166840.1332730.155753270−0.6039850.0560410.0865290.0980910.1201220.1170910.1261121−0.6244020.0817010.1229390.1288540.1404420.1504652−0.6470630.1458110.1360750.1766380.188541100−0.5754950.0113320.0709030.0257550.0981210.0403470.0798680.0820630.0819040.0852011−0.5804270.0105830.0780870.0627710.0814240.0653650.0984930.0876680.0960372−0.5981870.0436840.0699710.0950370.0771360.0973130.1063560.1086913−0.6039850.0560410.0865290.0980910.1201220.1170910.1261124−0.6244020.0817010.1229390.1288540.1404420.150465370−0.6109730.0714030.0647310.1272950.1173760.0979990.1321691−0.6355910.1010630.1212870.1327710.1236430.1568262−0.6352780.1237880.1594740.1571160.194901100−0.6081310.0678510.0404620.0261150.0872990.0693540.0831180.0793180.0633110.0913041−0.6014530.0545420.0500850.0453730.0946790.0991950.0820390.0755340.1000062−0.6050260.0604170.0609510.0638630.1264910.0947470.0851350.1134223−0.6109730.0714030.0647310.1272950.1173760.0979990.1321694−0.6355910.1010630.1212870.1327710.1236430.156826


## 3.1. BLUEs of Parameters

The PDF of the scale-parameter EPL distribution is(27)fy=δγυ2σ1+υyσγ−11+yσγ1−1+υ+υy/σγ1+υe−υy/σγδ−1e−υy/σγ,y>0,δ,γ,υ,σ>0,and the PDF of the location-scale parameter EPL distribution is(28)fy=δγυ2σ1+υy−μσγ−11+y−μσγe−υy−μ/σγ×1−1+υ+υy−μ/σγ1+υe−υy−μ/σγδ−1,y>0,δ,γ,υ,σ,μ>0.LetY1:m≤Y2:m≤⋯≤Yn−c:m, c=0,1,…,n−1 denote type-II right-censored sample from the location-scale parameter EPL distribution in equation (1). Let us denote Xr:m=Yr:m−μ/σ, EXr:m=μr;m1, 1≤r≤m−c, and CovXr:m,Xs:m=σr,s:m=μr,s:m1,1−μr:m1μs:m1, 1≤r<s≤m−c. Therefore,(29)Y=Y1:m,Y2:m,…,Ym−c:mT,μ=μ1:m,μ2:m,…,μm−c:mT,1=1,1,…,1T︸m−c,∑=σr,s,1≤r,s≤m−c,where μi:m=EYi:m, σii=VarYi:m and σij=CovYi:m,Yj:m, and i=1,2,…m−c. Then, the BLUEs of μ and σ can be computed as follows (see Arnold et al. [8]):(30)μ∗=∑r=1m−cprYr:m,σ∗=∑r=1m−cqrYr:m,where(31)pr=μT∑−1μ1T∑−1−μT∑−11μT∑−1μT∑−1μ1T∑−11−μT∑−112(32)qr=1T∑−11μT∑−1−1T∑−1μ1T∑−1μT∑−1μ1T∑−11−μT∑−112.The variances and covariance of these BLUEs can be computed as follows (see Arnold et al. [8]):(33)Varμ∗=σ2μT∑−1μμT∑−1μ1T∑−11−μT∑−112=σ2W1,Varσ∗=σ21T∑−11μT∑−1μ1T∑−11−μT∑−112=σ2W2,Covμ∗,σ∗=σ2−μT∑−11μT∑−1μ1T∑−11−μT∑−112=σ2W3.The values ofpr and qr are displayed in Tables 1 and 2 for different values of sample sizes n=7,10 and different censoring cases c=01n/2−1 and for some selected values for υ=1,2. The coefficient of the BLUEs pr and qr are given by (17) and (18), respectively, with conditions(34)∑i=1n−cpi=1,∑i=1n−cqi=0,which are used to check the computations accuracy.Table 1
Coefficients of the BLUEs of the location parameter.
υncpi, i=1,2,…,n−c1701.1367330.005506−0.023661−0.026954−0.024014−0.032131−0.03548211.161942−0.000791−0.040361−0.029379−0.039803−0.05160321.215986−0.044508−0.036811−0.061522−0.0731461001.092681−0.0146520.027777−0.023314−0.001847−0.019939−0.014359−0.014698−0.014324−0.01732511.105425−0.0051280.000896−0.0176790.001242−0.025909−0.023216−0.012611−0.02302221.129876−0.005171−0.017773−0.019798−0.008271−0.031966−0.019298−0.02760131.1367330.005506−0.023616−0.026954−0.024014−0.032131−0.03548241.161942−0.000796−0.040361−0.029379−0.039803−0.0516032701.1318010.004217−0.030454−0.006347−0.030267−0.030084−0.03886711.164716−0.015401−0.017184−0.042061−0.038315−0.05175521.204871−0.030066−0.038861−0.061012−0.0749331001.0780950.018066−0.0127110.010341−0.018799−0.013232−0.014746−0.014122−0.014181−0.01872211.0888180.021605−0.014117−0.007688−0.019094−0.010235−0.016854−0.018684−0.02369921.1071220.015291−0.019433−0.017916−0.008332−0.019141−0.029435−0.02815731.1318010.004217−0.030454−0.006347−0.030267−0.030084−0.03886741.164716−0.015401−0.017184−0.04206−0.038315−0.0517553701.1259910.003349−0.003324−0.026256−0.035498−0.02695−0.03731111.158534−0.006501−0.023706−0.036411−0.039839−0.05207821.203149−0.025636−0.048175−0.053915−0.0753871001.084513−0.0010820.012616−0.006894−0.0256820.004272−0.023087−0.010631−0.015161−0.01886511.0896080.0057370.014226−0.023515−0.010205−0.013899−0.022786−0.016716−0.02245121.113338−0.0069220.010884−0.026283−0.014407−0.026124−0.022595−0.02789231.1259910.003349−0.003324−0.026256−0.035498−0.026951−0.03731141.158534−0.006501−0.023706−0.036411−0.039839−0.052078Table 2
Coefficients of the BLUEs of the scale parameter.
υncqi, i=1,2,…,n−c170−0.6071910.0492650.1073570.0985630.1060760.1151310.1307981−0.6190960.0651740.1482120.1166840.1332730.1557532−0.6507290.1423760.1601710.1540250.194157100−0.6153920.0815140.0287930.0381890.0747110.0695230.0770510.0870840.0712460.0872831−0.6231590.0890410.0272650.0798450.0644780.0872260.0947280.0824620.0981152−0.6108790.0671920.0426660.1080530.0812350.1026640.0977070.1113633−0.6071910.0492650.1073570.0985630.1060760.1151310.1307984−0.6190960.0651740.1482120.1166840.1332730.155753270−0.6039850.0560410.0865290.0980910.1201220.1170910.1261121−0.6244020.0817010.1229390.1288540.1404420.1504652−0.6470630.1458110.1360750.1766380.188541100−0.5754950.0113320.0709030.0257550.0981210.0403470.0798680.0820630.0819040.0852011−0.5804270.0105830.0780870.0627710.0814240.0653650.0984930.0876680.0960372−0.5981870.0436840.0699710.0950370.0771360.0973130.1063560.1086913−0.6039850.0560410.0865290.0980910.1201220.1170910.1261124−0.6244020.0817010.1229390.1288540.1404420.150465370−0.6109730.0714030.0647310.1272950.1173760.0979990.1321691−0.6355910.1010630.1212870.1327710.1236430.1568262−0.6352780.1237880.1594740.1571160.194901100−0.6081310.0678510.0404620.0261150.0872990.0693540.0831180.0793180.0633110.0913041−0.6014530.0545420.0500850.0453730.0946790.0991950.0820390.0755340.1000062−0.6050260.0604170.0609510.0638630.1264910.0947470.0851350.1134223−0.6109730.0714030.0647310.1272950.1173760.0979990.1321694−0.6355910.1010630.1212870.1327710.1236430.156826


## 4. Approximate Inference

Here, we derive the1−α100% confidence intervals for the parameters μ and σ based on the following pivotal quantities:(35)U1=μ∗−μσW1,U2=σ∗−σσW2,U3=μ∗−μσ∗W1,where μ∗ and σ∗ are the BLUEs of μ and σ with variances σ2W1 and σ2W2, respectively. U1 is used to draw inference for μ when σ is known, while U3 can be used to draw inference for μ when σ is unknown. Similarly, U2 can be used to draw inference for σ when μ is unknown.The moments presented in Section2 are used to derive the confidence intervals of the location and scale parameters based on the pivotal quantities in equation (35).Hence,U1 and U2 can be rewritten as(36)U1=1W1∑i=1m−cpiXi:m=U1∗W1,U2=1W2∑i=1m−cqiXi:m−1=U2∗−1W2,where Xi:m=Yi:m−μ/σ, i=1,2,…,m−c, is the standardized form of the available type-II right-censored sample Yi:m, i=1,2,…,m−c.we consider finding the approximate distribution by using Edgeworth approximation for a statistic S (with mean 0 and variance 1) as(37)Hs≈Φs−ϕsα16s2−1+α2−324s3−3s+α172s5−10s2+15s,where α1 and α2 are the coefficients of skewness and kurtosis of S, respectively, and Φs and ϕs are the CDF and PDF of the standard normal distribution, respectively.


## 5. Numerical Aspects

The relations obtained in the preceding sections allow us to evaluate coefficients of the BLUEs. The coefficients of the BLUEs are presented in Tables1 and 2 while the variances and covariances of the BLUEs are presented in Table 3. From the results presented in Table 3, we observe that the variance of the BLUEs increases as the censoring level increases while the variance of the BLUEs decreases when the sample size increases and increases as υ increases. In addition, we observe that the covariances of the BLUEs decrease as the censoring level increases while the covariances of the BLUEs increase when the sample size increases and decrease as υ increases.Table 3
Variance and covariance of the BLUEs whenμ=0 and σ=1.
υncVar (μ)Var (σ)Cov (μ,σ)1700.0688210.135051−0.04126710.0933330.160655−0.05568320.1416130.201843−0.0837851000.0339130.088141−0.02011310.0420590.100032−0.02462320.0534050.116162−0.03197430.0688210.135051−0.04126740.0933330.160655−0.0556832700.0705620.135301−0.04110610.0983880.164369−0.05736220.1467010.208599−0.0856941000.0349450.089757−0.02055410.0423930.101301−0.02508220.0540410.114633−0.03089430.0705620.135301−0.04110640.0983880.164369−0.0573623700.0720620.137961−0.04164310.1008750.167103−0.05822920.1427070.211536−0.0849171000.0338310.091732−0.02027110.0421780.103774−0.02514120.0535430.119151−0.03173730.0720620.137961−0.04164340.1008750.167103−0.058229Tables4–6 display the simulated percentage points at υ=113 and sample size n=7,10. Table 7 presents the values of the mean, variance, and coefficient of skewness and kurtosis U1∗ and U2∗. The simulated average widths of confidence intervals are presented in Table 8. We observe that the Edgeworth approximations of the distributions of U1 and U2 both work quite satisfactorily; this is also clear from the average width of the confidence intervals based on U1 and U2 which are presented in Table 8.Table 4
Simulated values of the distribution ofU1 when μ=0 and σ=1.
υnc1%2.50%5%10%90%95%97.50%99%170−3.686292−3.686298−3.68631−1.234091.4488491.8113242.1909223.0492351−1.754612−1.459681−1.25981−1.035611.3076551.9249632.4997153.1594642−1.794954−1.508401−1.28934−1.051821.2984121.8951392.5055793.260451100−1.483964−1.326313−1.17203−0.997611.3492271.9389662.5498513.4140441−1.525359−1.336776−1.18298−0.99781.3430921.9145712.5402993.3592952−1.586863−1.376637−1.20636−1.014231.3381381.9224572.5203713.2325633−1.630238−1.425475−1.23309−1.017371.3110071.9172572.5171793.2259724−1.754612−1.459681−1.25981−1.035611.3076551.9249632.4997153.159464270−1.615403−1.391141−1.21902−1.016191.3199721.9226832.4914233.2528331−1.663496−1.448206−1.25977−1.032231.3328361.9184222.4949363.2073372−1.802659−1.515264−1.29324−1.055871.3190811.9074912.4483893.115657100−1.479384−1.299185−1.15835−0.984511.3436121.9840782.5482373.1822611−1.505824−1.348861−1.17004−0.995021.3359911.9552582.5337813.2051812−1.548062−1.357995−1.18398−1.000521.3460911.9505722.4960463.2537513−1.615403−1.391141−1.21902−1.016191.3199721.9226832.4914233.2528334−1.663496−1.448206−1.25977−1.032231.3328361.9184222.4949363.207337370−1.638623−1.422331−1.23184−1.012661.2807851.9039942.5159013.2685961−1.709657−1.464361−1.24497−1.018471.2769361.9239782.4995093.1921222−1.805743−1.519798−1.30221−1.052181.2996221.9094212.4571193.214435100−1.519226−1.329914−1.17265−1.000411.3240021.9371412.5412513.3265941−1.550921−1.350977−1.18953−1.006241.2937521.9122622.5548413.3738782−1.604826−1.395428−1.21302−1.01881.2946121.9215912.5120033.3765783−1.638623−1.422331−1.23184−1.012661.2807851.9039942.5159013.2685964−1.709657−1.464361−1.24497−1.018471.2769361.9239782.4995093.192122Table 5
Simulated values of the distribution ofU2 when μ=0 and σ=1.
υnc1%2.50%5%10%90%95%97.50%99%170−2.678889−2.471173−2.301446−2.0532130.4543720.9571411.4219791.9485981−2.554143−2.375446−2.212094−1.9747060.5264511.0176251.5042912.0777572−2.395787−2.253001−2.077632−1.8734220.6305711.1168811.6124112.259385100−3.019991−2.759098−2.548864−2.2881650.2371740.7209491.0859171.6047031−2.896556−2.664619−2.453476−2.1990370.3150510.8006661.1871911.7042992−2.807636−2.583198−2.374634−2.1205140.3927160.8534611.2992171.8129663−2.678889−2.471173−2.301446−2.0532130.4543720.9571411.4219791.9485984−2.554143−2.375446−2.212094−1.9747060.5264511.0176251.5042912.077757270−2.710056−2.505164−2.309474−2.0706790.4486020.9497311.3935451.9131121−2.582256−2.390288−2.218548−1.9674920.5238741.0320951.5089152.0872372−2.393321−2.230162−2.073431−1.8626780.6314681.1479241.6060342.213561100−3.024276−2.730335−2.541782−2.2707120.2751710.7347771.1508981.6654281−2.898164−2.655736−2.463116−2.2171350.3177730.8015811.2535011.7285122−2.823059−2.605918−2.400323−2.1435860.3691560.8638721.3250271.8258643−2.710056−2.505164−2.309474−2.0706790.4486020.9497311.3935451.9131124−2.582256−2.390288−2.218548−1.9674920.5238741.0320951.5089152.087237370−2.649813−2.466518−2.258944−2.0413990.4986090.9721351.4377021.9376011−2.519256−2.344106−2.173079−1.9469060.5682051.0956911.5563112.0722952−2.371208−2.198918−2.060219−1.8347120.6596111.1760771.6354012.248324100−2.926982−2.724922−2.529415−2.2550720.2661240.7458791.1682651.6876331−2.829318−2.645105−2.454084−2.1992880.3463920.8003951.2951531.7665732−2.748096−2.559353−2.350791−2.1237880.4117250.8955251.3608341.9095793−2.641983−2.466518−2.258944−2.0413990.4986090.9721351.4377021.9376014−2.519256−2.344106−2.173079−1.9469060.5682051.0956911.5563112.072295Table 6
Simulated values of the distribution ofU3 when μ=0 and σ=1.
υnc1%2.50%5%10%90%95%97.50%99%170−1.291825−1.262013−1.219325−1.1494672.2223953.5238474.9400927.1349441−1.293906−1.257118−1.216151−1.1405512.3503433.7650885.3893017.8523442−1.252756−1.219676−1.181444−1.1062552.5459324.1591476.0760958.899025100−1.323036−1.286828−1.242329−1.1653812.1100513.2236124.4983126.1674431−1.308061−1.270312−1.231131−1.1616112.1489753.2627314.5249376.6316672−1.294854−1.264969−1.227509−1.1574872.2529173.4117044.7047526.5013643−1.291825−1.262013−1.219325−1.1494672.2223953.5238474.9400927.1349444−1.293906−1.257118−1.216151−1.1405512.3503433.7650885.3893017.852344270−1.299635−1.254525−1.210859−1.1400812.3200863.5840045.1717397.0601321−1.269125−1.231214−1.188392−1.1193882.4772813.8719415.4052747.8981442−1.234267−1.201231−1.165376−1.0941192.6208754.1256885.9525718.910766100−1.316707−1.279803−1.234296−1.1556442.1277523.2572334.5117555.9759341−1.318328−1.274231−1.231598−1.1525252.1589893.3807084.5544986.3494532−1.306309−1.266361−1.220428−1.1427382.2270793.4869144.6948816.8761913−1.299635−1.254525−1.210859−1.1400812.3200863.5840045.1717397.0601324−1.269125−1.231214−1.188392−1.1193882.4772813.8719415.4052747.898144370−1.289032−1.249364−1.210882−1.1316582.2407143.4579874.9177267.2916641−1.258874−1.224641−1.180973−1.1030542.3198613.7130395.3049178.2160812−1.243127−1.210756−1.171678−1.0994422.5213284.0719256.0716649.205739100−1.340722−1.297027−1.250248−1.1740112.0807143.2467394.4594086.2772291−1.326197−1.285656−1.237163−1.1553452.0906443.2918854.7507226.4875072−1.313368−1.278006−1.229561−1.1521972.1731853.3662414.7640536.6506453−1.289032−1.249364−1.210882−1.1316582.2407143.4579874.9177267.2916644−1.258874−1.224641−1.180973−1.1030542.3198613.7130395.3049178.216081Table 7
Mean, variance, and coefficient of skewness and kurtosis ofU1∗ and U2∗ when μ=0 and σ=1.
υncU1U2MeanW1α1α2MeanW1α1α217000.0395181.2934152.911714−0.8698120.0775510.7424140.979527100.0535941.1925612.520105−0.7974930.0922520.7797771.008172200.0813181.1200992.226028−0.7114870.1159030.8556711.06398710000.0194741.4233283.072102−1.0766740.0506130.5777310.656743100.0241521.3905292.929123−1.0106570.0574410.6469920.849696200.0306661.3138592.779625−0.9378680.0667030.7043260.996806300.0395181.2934152.911714−0.8698120.0775510.7424140.979527400.0535941.1925612.520105−0.7974930.0922520.7797771.00817227000.0404461.3461192.959134−0.8722350.0775540.6573310.621787100.0563961.2905112.950683−0.7913610.0942150.7116860.724204200.0840881.1549962.724035−0.7024710.1195680.8457661.07662510000.0200311.5209313.976296−1.0709040.0514480.5638390.452005100.0242991.4646893.627171−1.0080420.0580650.5899490.502475200.0309761.4463713.457844−0.9476090.0657070.6258930.561158300.0404461.3461192.959134−0.8722350.0775540.6573310.621787400.0563961.2905112.950683−0.7913610.0942150.7116860.72420437000.0413361.4061693.581796−0.8624830.0791360.6784380.457074100.0578641.3383693.284062−0.7836760.0958530.7577210.647221200.0818591.1676492.806261−0.6965230.1213410.8704860.98131210000.0194061.4209983.140231−1.0577130.0526190.5634010.372104100.0241941.4683183.741814−0.9944510.0595260.5901850.356138200.0307131.3754743.216648−0.9280650.0683470.6318020.412006300.0413361.4061693.581796−0.8624830.0791360.6784380.457074400.0578641.3383693.284062−0.7836760.0958530.7577210.647221Table 8
Average width of the simulated CIs.
υncU1U2U390%95%90%95%90%95%1703.1416983.8825633.2592043.8987093.0186123.60634113.1781943.9431433.2506423.8992023.1814153.91041823.2007273.9636543.2213553.8361943.2747093.1016151003.1424293.8474223.2765593.8812343.4757813.90210313.1252973.8826413.2646983.9092363.9810263.15418523.1345473.8540413.2641953.9309443.5603453.92510133.1416983.8825633.2592043.8987093.6701653.25840443.1781943.9431433.2506423.8992023.4137243.1351692703.1503453.9426543.2585853.8931523.4361963.85318313.1847693.9593943.2297193.8797383.0665343.86047323.1844774.0139813.1945133.8654113.9496763.8666061003.1109973.8761643.2698133.8450163.4357063.87317913.0975453.8770753.2541423.8518813.8468343.10910323.1288143.8970083.2280953.8824153.4893363.95177533.1503453.9426543.2585853.8931523.3693413.78560843.1847693.9593943.2297193.8797383.3139533.1481383703.1358343.9382313.2310793.9042213.8891423.34887513.1689483.9638713.2687713.9004173.9719833.77114523.2116313.9769173.2362953.8343283.0971213.0968531003.1097863.8711663.2753293.8931873.4198193.80135413.1017883.9058173.2544813.9402583.8767753.31355923.1346113.9074313.2463163.9201873.4216463.91629733.1358343.9382313.2310793.9042213.6015933.06564443.1689483.9638713.2687713.9004173.2424893.029839


## 6. Real Data Analysis

In this section, we analyze two real data sets to show the importance of the proposed estimators. One data set from environmental monitoring and another data is from maximum flood level for the Susquehanna River at Harrisburg.Example 1.
Analysis of clean upgradient ground-water monitoring wells subjected to mg/L.In the first data set, we consider vinyl chloride data obtained from clean upgradient monitoring wells which was studied by many authors such as Bhaumik and Gibbons [26]; Krishnamoorthy et al. [27]; Bhaumik et al. [28]; and Kumar and Goyal [22, 23]. The data are as follows:(38)5.11.21.30.60.52.40.51.18.00.80.40.60.90.42.00.55.33.22.72.92.52.31.00.20.10.11.80.92.04.06.81.20.40.2.
Now, a random sample of size 10 is selected from the given data set 1.0, 1.2, 3.2, 2.4, 0.8, 2.0, 0.4, 0.2, 2.9, 1.2. By the EPL distribution for the given sample, we have obtained the maximum likelihood estimate ofδML=0.59023, γML=1.808770, and υML=0.434371. By computation, the Kolmogorov–Smirnov (K–S) distance and the corresponding p values are 0.163772 and 0.9513, respectively, which implies that the EPL distribution provides a reasonable model for this data. Moveover, the empirical cumulative distribution function (ECDF) plot and the Quantile-Quantile (Q-Q) plots are also presented under the vinyl chloride data as shown in Figure 1, which also imply that the EPL distribution can be used as a proper model to fit this data.
Then, by using the BLUE coefficients in Tables1 and 2, we have(39)μ∗=∑j=1npjXj:n=0.024467,σ∗=∑j=1nqjXj:n=0.992854.Figure 1
QQ plot and CDF of the real data set I on the EPL distribution. (a) QQ plot of sample vs theoretical distribution; alpha = 1, beta = 1.1, and theta = 1. (b) CDF for sample and theoretical CDF.
(a)(b)Example 2.
Analysis of the maximum flood level for the Susquehanna River at Harrisburg, Pennsylvania.The second data set presents the maximum flood level for the Susquehanna River at Harrisburg, Pennsylvania, and it was studied by Dumonceaux and Antle [29]. The data are(40)0.6540.6130.3150.4490.2970.4020.3790.4230.3790.32350.2690.7400.4180.4120.4940.4160.3380.3920.4840.265.
Now, a random sample of size 10 is selected from the given data set 0.418, 0.265, 0.613, 0.269, 0.654, 0.338, 0.315, 0.379, 0.297, 0.412. By the EPL distribution for the given sample, we have obtained the maximum likelihood estimate ofδML=3479.9773640, γML=0.484329, and υML=14.626781. By computation, the Kolmogorov–Smirnov (K–S) distance and the corresponding p values are 0.14282 and 0.9694, respectively, which implies that the EPL distribution provides a reasonable model for this data. Moreover, as a further illustration, the empirical cumulative distributions plot and overlay the theoretical EPL distribution, the quantile-quantile (Q-Q) plots are also presented under the maximum flood level data, as shown in Figure 2, which also imply that the EPL distribution can be used as a proper model to fit this data.
Then, by using the BLUE coefficients in Tables1 and 2, we have(41)μ∗=∑j=1npjXj:n=0.246173,σ∗=∑j=1nqjXj:n=0.099934.Figure 2
QQ plot and CDF of the real data set II on the EPL distribution.
(a)(b)


## 7. Conclusion

In this paper, we have considered the EPL distribution when data are available in the form of order statistics. We first presented expressions for the single, double, triple, and quadruple moments. By using these moments, we have calculated the BLUEs of the location and scale parameters and the coefficient of skewness and kurtosis for some linear pivotal quantities. In the simulation study, we observed that the variance of the BLUEs increases when a high censoring level is taken into account; however, it decreases when a large value of parameterυ is considered corresponding to a fixed value of parameters δ and γ. Also, the covariance of the BLUEs decreases as the censoring level increases while it increases when the sample size increases and it decreases as υ increases. We next considered the distributions of the pivotal quantities in terms of Edgeworth approximation. These pivotal quantities are used to construct the interval estimation for the location and scale parameters. From our finding, we see that the moments of order statistics of the distribution are well behaved. This will encourage the study of the other properties of order statistics for a future research.



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*Source: 4918342-2020-09-23.xml*