# Inferences for Generalized Pareto Distribution Based on Progressive First-Failure Censoring Scheme

Authors: Rashad M. El-Sagheer; Taghreed M. Jawa; Neveen Sayed-Ahmed
Journal: Complexity
 (2021)

---

## Abstract

In this article, we consider estimation of the parameters of a generalized Pareto distribution and some lifetime indices such as those relating to reliability and hazard rate functions when the failure data are progressive first-failure censored. Both classical and Bayesian techniques are obtained. In the Bayesian framework, the point estimations of unknown parameters under both symmetric and asymmetric loss functions are discussed, after having been estimated using the conjugate gamma and discrete priors for the shape and scale parameters, respectively. In addition, both exact and approximate confidence intervals as well as the exact confidence region for the estimators are constructed. A practical example using a simulated data set is analyzed. Finally, the performance of Bayes estimates is compared with that of maximum likelihood estimates through a Monte Carlo simulation study.


---

## Body

## 1. Introduction

In life testing and reliability analysis, some units can be lost or withdrawn from the experiment before failure occurs. One of the major reasons for removal of the experimental units is to save the working experimental units for future use, thereby conserving the cost and time associated with testing. This leads us to use the censoring schemes. The type-II censoring can be considered a common type of censored scheme. Many authors have studied the statistical inference for different probability distributions using progressive type-II censoring, including Balakrishnan and Sandhu [1, 2], Cohen [3], Mann [4], Ng [5], Balakrishnan et al. [6], Gibbons and Vance [7], Yuen and Tse [8], Ng et al. [9], Balakrishnan [10], Soliman [11, 12], Madi and Raqab [13], Mahmoud et al. [14], Mahmoud et al. [15], Soliman et al. [16], El-Sagheer [17–19], Mahmoud et al. [20], El-Sagheer and Hasaballah [21], El-Sagheer et al. [22], and Soliman et al. [23]. Recently, Zhang and Gui [24] studied the statistical inference for the lifetime performance index of Pareto distribution based on progressive type-II censored sample.On the other hand, Viveros and Balakrishnan [25] have described a life test in which the experimenter can decide to divide the items being tested into several groups and then run all the items at the same time until occurrence of the first failure in each group. Such a censoring scheme is called first-failure censoring. For more details about statistical inference using first-failure censoring, it is recommended that the reader refers to Wu and Yu [26], Wu et al. [27], Lee et al. [28], and Wu et al. [29]. However, using this censoring scheme does not enable the experimenter to remove experimental units from the test until the first failure is observed. For this reason, Wu and Kuş [30] introduced a life testing scheme, which combines first-failure censoring with a progressive type-II censoring called a progressive first-failure censoring (Pro-F-F-C) scheme. Many previous studies have discussed inference under a Pro-F-F-C scheme for different lifetime distributions, for example, Weibull by Wu and Kuş [30], Burr Type XII by Soliman et al. [31, 32], Gompertz by Soliman et al. [33], Lomax by Mahmoud et al. [34], Compound Rayleigh by Abushal [35], Generalized Inverted Exponential by Ahmed [36], the Mixture of Weibull and Lomax by Mahmoud et al. [37], and exponentiated Frechet by Soliman et al. [38]. Recently, Cai and Gui [39] discussed the classical and Bayesian inference for a Pro-F-F-C left-truncated normal distribution.Generalized Pareto distribution (GPD) is a significant continuous lifetime distribution. It plays a key role in statistical inference studies and reliability problems. It is also well known for being a distribution that has decreasing failure rate property. The pdf and cdf of a random variableX have a GPD given, respectively, as(1)fXx;α,β=αβαx+β−α+1,x>0,α,β>0,(2)FXx;α,β=1−βαx+β−α,x>0,α,β>0,where α and β are the shape and scale parameters, respectively. The survival and hazard rate functions of GPD at mission time t are given by the following expressions:(3)st=βαt+β−α,t>0,(4)ht=αt+β−1,t>0.For more details about GPD, its properties, and applications, see Kremer [40]. In this article, we obtain the Bayes estimates and MLEs for the unknown quantities of the GPD using a Pro-F-F-C scheme. The approximate confidence intervals (ACIs) for α and β are constructed based on the asymptotic normality of MLEs. In the Bayesian framework, the point estimates of unknown parameters under squared error (SE), linear-exponential (LINEX), and general entropy (GE) showing loss functions are discussed. The process is done using the conjugate gamma prior for the shape parameter and discrete prior for the scale parameter β. The exact confidence interval and exact confidence region for the estimators are then derived. To evaluate and compare the performance of these proposed inference procedures, a simulation study with different parameter values is undertaken. Additionally, a numerical example using simulated data set is studied to show the practicality and usefulness of these proposed methods.The rest of the paper is arranged as follows. Section2 deals with the classical method of estimation. Bayes estimators relative to different loss functions are considered in Section 3. In Section 4, the ACIs, exact confidence intervals, and exact confidence regions for the parameters are discussed. In Section 5, the proposed procedures obtained in the previous sections are investigated using simulated data. A simulation study is conducted to compare the proposed procedures in Section 6. Finally, a conclusion is provided in Section 7.


## 2. Maximum Likelihood Estimation

Letxi=xi:m:n:kR, i=1,2,…,m, be a Pro-F-F-C order statistics from the GPD with the progressive censoring scheme R=R1,R2,…,Rm. According to Wu and Kuş [30], the joint probability density function can be written as(5)f1,2,…,mx1:m:n:kR,…,xm:m:n:kR∝km∐j=1mfxj:m:n:kR1−Fxj:m:n:kRkRj+1−1.From (1), (2), and (5), the likelihood function Lx¯;α,β is given by(6)Lx¯;α,β∝kmαm∏i=1mβαkRi+1xi+β−αkRi+1+1.Thus, the log-likelihood functionℓx¯;α,β is(7)ℓx¯;α,β∝mlogk+mlogα+∑i=1mαkRi+1logβ−∑i=1mαkRi+1+1logxi+β.By equating each result of the first-order derivatives of log-likelihood function with respect toα and β, to zero, we obtain(8)ℓx¯;α,β∂α=mα+∑i=1mkRi+1logβ−∑i=1mkRi+1logxi+β=0,(9)ℓx¯;α,β∂β=∑i=1mαkRi+1β−∑i=1mαkRi+1+1xi+β=0.Hence,(10)α^=m∑i=1mkRi+1logxi+β^−∑i=1mkRi+1logβ^−1,and β^ the solution of(11)∑i=1mα^kRi+1β^−∑i=1mα^kRi+1+1xi+β^=0.Since there is no closed form of the solution to the above equations, the Newton–Raphson method (NRM) is widely used to obtain the desired MLEs in such situations. Once MLEs ofα and β are obtained, the MLEs of st and ht for given t can be obtained by the invariant property of the MLEs as(12)s^t=β^α^t+β^−α^,t>0,(13)h^t=α^t+β^−1,t>0.


## 3. Bayesian Estimation

Bayes estimation is quite different from the MLE method because it takes into consideration both the information from observed sample data and the prior information. Bayes’ theorem is completely dependent on the parameter estimation through calculation of the posterior distribution. As calculating the posterior distribution is conditional on the data, this requires explicit specification of the prior distribution model parameters. Furthermore, in order to gain the best estimate of the unknown parameter, it is necessary to determine the appropriate loss functions.The next step is to take into account different loss functions. First, we consider the square error (SE) loss function which is widely used in the literature. Because of the symmetry nature of this function, it gives equal weight to overestimation as well as underestimation. Under SE, the Bayesian estimate (BE) of any function of parameters, sayψΘ=uα,β,s,h, is the unconditional posterior mean which is given as(14)ψ^BSΘ=EψΘ=∫ΘψΘπ∗ΘdΘ.However, in many situations, the parameter may be overestimated or show serious consequences of underestimation, or vice versa. In such cases, an asymmetric loss function, which associates greater importance to overestimation or underestimation, can be taken into consideration for parameters estimation. A beneficial asymmetric loss function is the LINEX loss as follows:(15)LLINEXψ^Θ,ψΘ=eaψ^Θ−ψΘ−aψ^Θ−ψΘ−1,where a is a shape parameter whose sign refers to the direction and its magnitude represents the degree of symmetry. Moreover, for a figure close to zero, the LINEX loss more or less becomes a SE loss. Thus, the BE of ψΘ under this loss function is given by(16)ψ^BLΘ=−1alogEe−aψΘ=−1alog∫Θe−aψΘπ∗ΘdΘ.Next, we consider the GE loss function as follows:(17)LGEψ^Θ,ψΘ=ψ^ΘψΘq−qlogψ^ΘψΘ−1,where q is a shape parameter which represents departure from symmetry. Subsequently, based on the GE loss functions, the BE of ψΘ is obtained as(18)ψ^BGΘ=EψΘ−q−1/q=∫ΘψΘ−qπ∗ΘdΘ−1/q.It is remarked that forq=−1, the BE of ψΘ concurs with the BE under SE loss function.


### 3.1. Posterior Analysis

In this subsection, we consider that the parameterβ a discrete prior and α has a conjugate gamma prior. Suppose that β=βj, j=1,2,…,N, then(19)πβ=Prβ=βj=ηj,where 0≤ηj≤1 and ∑j=1Nηj=1. Further, α has(20)πα|β=βj=ajbjΓbjαbj−1exp−ajα,α;aj,bj>0.Then, the posterior distribution ofα takes the form as follows:(21)π∗α|β=βj;Tj=Tj+ajbj+mΓbj+mαbj+m−1exp−αTj+aj,where(22)Tj=∑i=1mkRi+1logxi+βj−logβj.The joint posterior ofα and βj using (6), (19), and (20) is(23)π∗α,β=βj,Tj=ajbjvjηjk2Γbjαbj+m−1exp−αTj+aj,where(24)k2=∑j=1NajbjvjηjΓbj+mΓbjTj+ajbj+m,vj=∏i=1mxi+βj−1.By using the Bayes theorem for discrete variables, the marginal posterior probability ofβ is(25)Pj=Prβ=βj|Tj=ajbjvjηjΓbj+mk2ΓbjTj+ajbj+m,where k2 and vj are given in (24); the marginal posterior probability of α is(26)π∗α|Tj=∑j=1mπ∗α|β=βj,Tj.


### 3.2. BE under SE Loss

In this subsection, we obtain the BE ofα, β, st, and ht under SE loss function. By using (14), (21), and (25), the BEs α˜BS, β˜BS, s˜BSt, and h˜BSt are given by(27)α˜BS=∫0∞∑j=1NαPjπ∗α|β=βj,Tjdα=∑j=1NPjbj+mTj+aj,β˜BS=Eββ|x¯=∑j=1NβjPj,s˜BSt=∑j=1NPj1+log1+t/βjTj+ajbj+m,(28)h˜BSt=∑j=1NPjbj+mt+βjTj+aj.


### 3.3. BE under LINEX Loss

Based on (16), (21), and (25), the BEs β˜BL, α˜BL, s˜BLt, and h˜BLt are(29)β˜BL=−1alog∑j=1NPjexp−aβj,α˜BL=−1alog∑j=1NPj1+aTj+aj−bj+m,s˜BLt=−1alog∑j=1N∑ε=1∞−aεε!Pj1+εlog1+t/βjTj+aj−bj+m,(30)h˜BLt=−1alog∑j=1NPj1+at+βjTj+aj−bj+m.


### 3.4. BE under GE Loss

From (18), (21), and (25), the BEs β˜BG, α˜BG, s˜BGt, and h˜BGt are, respectively,(31)β˜BG=∑j=1Nβj−qPj−1/q,α˜BG=∑j=1NPjTj+ajqΓbj+m−qΓbj+m−1/q,s˜BGt=∑j=1NPj1−qlog1+t/βjTj+aj−bj+m−1/q,(32)h˜BGt=∑j=1NPjt+βjq×Tj+ajqΓbj+m−qΓbj+m−1/q.To perform the calculations in these subsections, the values ofaj and bj must be found in (20). We use the prior expectation of st conditional on β=βj. Thus, from (3) and (20), we get(33)Est|βj=1+log1+t/βjajbj.


## 3.1. Posterior Analysis

In this subsection, we consider that the parameterβ a discrete prior and α has a conjugate gamma prior. Suppose that β=βj, j=1,2,…,N, then(19)πβ=Prβ=βj=ηj,where 0≤ηj≤1 and ∑j=1Nηj=1. Further, α has(20)πα|β=βj=ajbjΓbjαbj−1exp−ajα,α;aj,bj>0.Then, the posterior distribution ofα takes the form as follows:(21)π∗α|β=βj;Tj=Tj+ajbj+mΓbj+mαbj+m−1exp−αTj+aj,where(22)Tj=∑i=1mkRi+1logxi+βj−logβj.The joint posterior ofα and βj using (6), (19), and (20) is(23)π∗α,β=βj,Tj=ajbjvjηjk2Γbjαbj+m−1exp−αTj+aj,where(24)k2=∑j=1NajbjvjηjΓbj+mΓbjTj+ajbj+m,vj=∏i=1mxi+βj−1.By using the Bayes theorem for discrete variables, the marginal posterior probability ofβ is(25)Pj=Prβ=βj|Tj=ajbjvjηjΓbj+mk2ΓbjTj+ajbj+m,where k2 and vj are given in (24); the marginal posterior probability of α is(26)π∗α|Tj=∑j=1mπ∗α|β=βj,Tj.


## 3.2. BE under SE Loss

In this subsection, we obtain the BE ofα, β, st, and ht under SE loss function. By using (14), (21), and (25), the BEs α˜BS, β˜BS, s˜BSt, and h˜BSt are given by(27)α˜BS=∫0∞∑j=1NαPjπ∗α|β=βj,Tjdα=∑j=1NPjbj+mTj+aj,β˜BS=Eββ|x¯=∑j=1NβjPj,s˜BSt=∑j=1NPj1+log1+t/βjTj+ajbj+m,(28)h˜BSt=∑j=1NPjbj+mt+βjTj+aj.


## 3.3. BE under LINEX Loss

Based on (16), (21), and (25), the BEs β˜BL, α˜BL, s˜BLt, and h˜BLt are(29)β˜BL=−1alog∑j=1NPjexp−aβj,α˜BL=−1alog∑j=1NPj1+aTj+aj−bj+m,s˜BLt=−1alog∑j=1N∑ε=1∞−aεε!Pj1+εlog1+t/βjTj+aj−bj+m,(30)h˜BLt=−1alog∑j=1NPj1+at+βjTj+aj−bj+m.


## 3.4. BE under GE Loss

From (18), (21), and (25), the BEs β˜BG, α˜BG, s˜BGt, and h˜BGt are, respectively,(31)β˜BG=∑j=1Nβj−qPj−1/q,α˜BG=∑j=1NPjTj+ajqΓbj+m−qΓbj+m−1/q,s˜BGt=∑j=1NPj1−qlog1+t/βjTj+aj−bj+m−1/q,(32)h˜BGt=∑j=1NPjt+βjq×Tj+ajqΓbj+m−qΓbj+m−1/q.To perform the calculations in these subsections, the values ofaj and bj must be found in (20). We use the prior expectation of st conditional on β=βj. Thus, from (3) and (20), we get(33)Est|βj=1+log1+t/βjajbj.


## 4. Interval Estimation

This section deals with ACIs, exact CIs, and exact confidence regions for the parametersα and β of GPD based on Pro-F-F-C.


### 4.1. Asymptotic Confidence Intervals

The asymptotic normality of the MLEs can be used to construct ACIs for parametersα and β by using Fisher information matrix (FIM). The FIM can be written as I=Iij where(34)Iij=E−∂2ℓΦ∂ϕi∂ϕj,i,j=1,2,where Φ=ϕ1,ϕ2=α,β. The asymptotic variance-covariance matrix of the parameters α and β can be obtained by inverting the observed FIM Iij as follows:(35)I−1α^,β^=−∂2ℓ∂α2−∂2ℓ∂α∂β−∂2ℓ∂β∂α−∂2ℓ∂β2α^,β^−1=varα^covα^,β^covβ^,α^varβ^,with(36)∂2ℓ∂α2=−mα2,∂2ℓ∂α∂β=∂2ℓ∂β∂α=∑i=1mkRi+1β−∑i=1mkRi+1xi+β,(37)∂2ℓ∂β2=∑i=1mαkRi+1+1xi+β2−∑i=1mαkRi+1β2.Thus,(38)α^,β^∼Nα,β,I0−1α^,β^.The1−δ100% ACIs for α and β become(39)α^−Zδ/2varα^,α^+Zδ/2varα^,β^−Zδ/2varβ^,β^+Zδ/2varβ^,where zδ is 1001−δth upper percentile of standard normal variate N0,1.


### 4.2. Exact Confidence Intervals

Letx1:m:n:kR<x2:m:n:kR<⋯<xm:m:n:kR denote a Pro-F-F-C sample from GPD with parameters α and β, and let(40)Ui:m:n:kR=kαlog1+xi:m:n:kRβ,i=1,2,…,m.It is remarked thatU1:m:n:kR<U2:m:n:kR<…<Um:m:n:kR is a progressively censored sample of exponential distribution (ED) with mean 1. Let us assume the following:(41)W1=nU1:m:n:kRW2=n−R1−1U2:m:n:kR−U1:m:n:kRW3=n−R1−R2−2U3:m:n:kR−U2:m:n:kR⋮Wm=n−R1−…−Rm−1−m+1Um:m:n:kR−Um−1:m:n:kR.According to Thomas and Wilson [41], the generalized spacings W1,W2,…,Wm are iid as standard ED; hence,(42)ζj=2∑i=1jWi,has X22j, and(43)ψj=2∑i=j+1mWi,has X22m−j. To construct the confidence intervals for α and β, we consider pivotal quantities:(44)ξj=ψj/2m−jζj/2j=jm−j.2∑i=j+1mWi2∑i=1jWi=jm−j.R1+R2+⋯+Rj+j−n+∑i=j+1mRi+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/βn−R1−R2−…−Rj−1−j+1+∑i=1j−1Ri+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/β,j=1,2,…,m−1,(45)η=ψj+ζj=2∑i=1mWi=2∑i=1mRi+1Ui:m:n:kR=2kα∑i=1mRi+1log1+xi:m:n:kRβ.It can be easily shown thatξj∼F2m−j,2j where j=1,2,…,m−1, m>1, and η∼X22m. Also, ξj and η are independent. To construct an exact confidence interval for β and exact joint confidence region for β and α, we need to analyze the following two lemmas.Lemma 1.
For any positive real numbersb>a>0, qγ=ln1+bγ/ln1+aγ is a strictly increasing function of γ, where γ>0.Lemma 2.
For a given set of observations0<x1:m:n:kR<x2:m:n:kR<…<xm:m:n:kR<∞, the function ξj is a strictly increasing function of β when β>0. Furthermore,(I)
Forxm−1:m:n:kR≤1, there is a unique solution for the given equation ξj=t, where t>0.(II)
Letx0:m:n:kR=0. For xl:m:n:kR≤1<xl+1:m:n:kR, there is a unique solution for the given equation ξj=t where(46)0<t<jm−j.∑i=j+1mRi+1logxi:m:n:kR−n−R1−R2−…−Rj−jlogxj:m:n:kRn−R1−R2−…−Rj−1−j+1logxj:m:n:kR+∑i=l+1j−1Ri+1logxi:m:n:kR,forl=0,1,…,j−1 and j=1,2,…,m−1. Using the same arguments and notations in Wu et al. [42], Lemma 1 and Lemma 2 can be proved.


### 4.3. Exact Confidence Interval forβ

Suppose thatxi,m,n,kR, i=1,2,…,m, denote a Pro-F-F-C sample from GPD α,β, with censoring scheme R1,R2,…,Rm. For any 0<δ<1, a1001−δ% confidence interval for β is as follows. We know that ξj∼F2m−j,2j by Lemma1 and Lemma 2ξj strictly increases in β when β>0, where(1)
Forxm−1:m:n:kR≤1, there is a unique solution for the given equation ξj=t, where t>0.(2)
Letx0:m:n:kR=0. For xl:m:n:kR≤1<xl+1:m:n:kR, there is a unique solution for the equation ξj=t.Hence, for0<δ<1, from (44), we obtain(47)F1−δ22m−j,2j<ξj<Fδ22m−j,2j.Thus, a1001−δ% confidence interval for β is(48)ΦXR,F1−δ/22m−j,2j<β<ΦXR,Fδ/22m−j,2j,where XR=X1:m:n:kR,X2:m:n:kR,…,Xm:m:n:kR and ΦXR,t is the solution for β for the equation:(49)R1+R2+⋯+Rj+j−n+∑i=j+1mRi+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/βn−R1−R2−…−Rj−1−j+1+∑i=1j−1Ri+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/β=tm−jj.


### 4.4. Exact Confidence Region forβ and α

By the same way, from (45), it is clear that(50)η=2kα∑i=1mRi+1log1+xi:m:n:kRβ,where η∼X22m. For 0<δ<1, we have(51)PF1+1−δ/2/22m−j,2j<ξj<F1−1−δ/22m−j,2j=1−δ,(52)Pχ1+1−δ/22m2<η<χ1−1−δ/22m2=1−δ.Then, we obtain(53)PF1+1−δ/22m−j,2j<ξj<F1−1−δ/22m−j,2j,χ1+1−δ/22m2<η<χ1+1−δ/22m2=1−δ.This is equivalent to(54)PΦXR,F1+1−δ/22m−j,2j<β<ΦXR,F1−1−δ/22m−j,2j,χ1+1−δ/22m22k∑i=1mRi+1log1+xi:m:n:kR/β<α<χ1−1−δ/22m22k∑i=1mRi+1log1+xi:m:n:kR/β=1−δ.


## 4.1. Asymptotic Confidence Intervals

The asymptotic normality of the MLEs can be used to construct ACIs for parametersα and β by using Fisher information matrix (FIM). The FIM can be written as I=Iij where(34)Iij=E−∂2ℓΦ∂ϕi∂ϕj,i,j=1,2,where Φ=ϕ1,ϕ2=α,β. The asymptotic variance-covariance matrix of the parameters α and β can be obtained by inverting the observed FIM Iij as follows:(35)I−1α^,β^=−∂2ℓ∂α2−∂2ℓ∂α∂β−∂2ℓ∂β∂α−∂2ℓ∂β2α^,β^−1=varα^covα^,β^covβ^,α^varβ^,with(36)∂2ℓ∂α2=−mα2,∂2ℓ∂α∂β=∂2ℓ∂β∂α=∑i=1mkRi+1β−∑i=1mkRi+1xi+β,(37)∂2ℓ∂β2=∑i=1mαkRi+1+1xi+β2−∑i=1mαkRi+1β2.Thus,(38)α^,β^∼Nα,β,I0−1α^,β^.The1−δ100% ACIs for α and β become(39)α^−Zδ/2varα^,α^+Zδ/2varα^,β^−Zδ/2varβ^,β^+Zδ/2varβ^,where zδ is 1001−δth upper percentile of standard normal variate N0,1.


## 4.2. Exact Confidence Intervals

Letx1:m:n:kR<x2:m:n:kR<⋯<xm:m:n:kR denote a Pro-F-F-C sample from GPD with parameters α and β, and let(40)Ui:m:n:kR=kαlog1+xi:m:n:kRβ,i=1,2,…,m.It is remarked thatU1:m:n:kR<U2:m:n:kR<…<Um:m:n:kR is a progressively censored sample of exponential distribution (ED) with mean 1. Let us assume the following:(41)W1=nU1:m:n:kRW2=n−R1−1U2:m:n:kR−U1:m:n:kRW3=n−R1−R2−2U3:m:n:kR−U2:m:n:kR⋮Wm=n−R1−…−Rm−1−m+1Um:m:n:kR−Um−1:m:n:kR.According to Thomas and Wilson [41], the generalized spacings W1,W2,…,Wm are iid as standard ED; hence,(42)ζj=2∑i=1jWi,has X22j, and(43)ψj=2∑i=j+1mWi,has X22m−j. To construct the confidence intervals for α and β, we consider pivotal quantities:(44)ξj=ψj/2m−jζj/2j=jm−j.2∑i=j+1mWi2∑i=1jWi=jm−j.R1+R2+⋯+Rj+j−n+∑i=j+1mRi+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/βn−R1−R2−…−Rj−1−j+1+∑i=1j−1Ri+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/β,j=1,2,…,m−1,(45)η=ψj+ζj=2∑i=1mWi=2∑i=1mRi+1Ui:m:n:kR=2kα∑i=1mRi+1log1+xi:m:n:kRβ.It can be easily shown thatξj∼F2m−j,2j where j=1,2,…,m−1, m>1, and η∼X22m. Also, ξj and η are independent. To construct an exact confidence interval for β and exact joint confidence region for β and α, we need to analyze the following two lemmas.Lemma 1.
For any positive real numbersb>a>0, qγ=ln1+bγ/ln1+aγ is a strictly increasing function of γ, where γ>0.Lemma 2.
For a given set of observations0<x1:m:n:kR<x2:m:n:kR<…<xm:m:n:kR<∞, the function ξj is a strictly increasing function of β when β>0. Furthermore,(I)
Forxm−1:m:n:kR≤1, there is a unique solution for the given equation ξj=t, where t>0.(II)
Letx0:m:n:kR=0. For xl:m:n:kR≤1<xl+1:m:n:kR, there is a unique solution for the given equation ξj=t where(46)0<t<jm−j.∑i=j+1mRi+1logxi:m:n:kR−n−R1−R2−…−Rj−jlogxj:m:n:kRn−R1−R2−…−Rj−1−j+1logxj:m:n:kR+∑i=l+1j−1Ri+1logxi:m:n:kR,forl=0,1,…,j−1 and j=1,2,…,m−1. Using the same arguments and notations in Wu et al. [42], Lemma 1 and Lemma 2 can be proved.


## 4.3. Exact Confidence Interval forβ

Suppose thatxi,m,n,kR, i=1,2,…,m, denote a Pro-F-F-C sample from GPD α,β, with censoring scheme R1,R2,…,Rm. For any 0<δ<1, a1001−δ% confidence interval for β is as follows. We know that ξj∼F2m−j,2j by Lemma1 and Lemma 2ξj strictly increases in β when β>0, where(1)
Forxm−1:m:n:kR≤1, there is a unique solution for the given equation ξj=t, where t>0.(2)
Letx0:m:n:kR=0. For xl:m:n:kR≤1<xl+1:m:n:kR, there is a unique solution for the equation ξj=t.Hence, for0<δ<1, from (44), we obtain(47)F1−δ22m−j,2j<ξj<Fδ22m−j,2j.Thus, a1001−δ% confidence interval for β is(48)ΦXR,F1−δ/22m−j,2j<β<ΦXR,Fδ/22m−j,2j,where XR=X1:m:n:kR,X2:m:n:kR,…,Xm:m:n:kR and ΦXR,t is the solution for β for the equation:(49)R1+R2+⋯+Rj+j−n+∑i=j+1mRi+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/βn−R1−R2−…−Rj−1−j+1+∑i=1j−1Ri+1log1+xi:m:n:kR/β/log1+xj:m:n:kR/β=tm−jj.


## 4.4. Exact Confidence Region forβ and α

By the same way, from (45), it is clear that(50)η=2kα∑i=1mRi+1log1+xi:m:n:kRβ,where η∼X22m. For 0<δ<1, we have(51)PF1+1−δ/2/22m−j,2j<ξj<F1−1−δ/22m−j,2j=1−δ,(52)Pχ1+1−δ/22m2<η<χ1−1−δ/22m2=1−δ.Then, we obtain(53)PF1+1−δ/22m−j,2j<ξj<F1−1−δ/22m−j,2j,χ1+1−δ/22m2<η<χ1+1−δ/22m2=1−δ.This is equivalent to(54)PΦXR,F1+1−δ/22m−j,2j<β<ΦXR,F1−1−δ/22m−j,2j,χ1+1−δ/22m22k∑i=1mRi+1log1+xi:m:n:kR/β<α<χ1−1−δ/22m22k∑i=1mRi+1log1+xi:m:n:kR/β=1−δ.


## 5. Numerical Computations

Consider a Pro-F-F-C sample generated from GPD showingα=0.3 and β=1.5. The data consist of 120 observations, grouped into n=30 sets, with 4 items within each group k=4. The Pro-F-F-C sample of size 10 out of 30 groups with the corresponding censoring scheme R is given in Table 1. The MLEs of α and β using NRM are computed, and then both st and ht are calculated at t=0.451.Table 1
Simulated Pro-F-F-C.
i12345678910Ri10011511100xiR0.07810.15820.16940.20400.30660.49090.89121.07054.81114.123To compute the BEs, we first estimate two values ofst using a nonparametric procedure sti=xi,m,n,kR=m−i+0.625/m+0.25, i=1,2,…,m. Using the available data, we obtained st1=0.1694=0.7439 and st2=4.8110=0.1585. These two priors are substituted into (33), where aj and bj are obtained numerically for each given βj, and ηj, j=1,2,…,10, using the NRM. Table 2 displays the values of aj, bj, and Pj for each given βj and ηj. The results of MLE and BE for α, β, st, and ht are presented in Table 3. By using (45), the 95% ACIs of α and β are 0,0.7068 and 0,4.4543. For j=2, we need the percentiles F0.02518,2=0.2193 and F0.97518,2=39.4424 to construct the 95% CI for β. According to (44), the 95% exact confidence interval of β is calculated as 0.2193,7.9943. For the given F0.012718,2=0.1780, F0.987318,2=78.1835, χ0.0127202=8.5737, and χ0.9873202=36.7141, the 95% joint confidence region for β and α is(55)0.2193<β<7099438.57378∑i=1mRi+1log1+xi:m:n:kR/β<α<36.71418∑i=1mRi+1log1+xi:m:n:kR/β.Table 2
The hyperparameter values.
j12345678910ηj0.10.10.10.10.10.10.10.10.10.1βj1.01.11.21.31.41.51.61.71.81.9aj0.69950.59660.51840.45710.40870.36840.33490.30660.28240.2615bj1.46471.37431.30541.24571.19761.15741.12371.09381.06641.0434Pj0.10590.10580.10850.10520.10410.10230.09970.09610.09210.0857Table 3
The MLEs and BEs ofα,β,st, and ht where s0.450=0.9243 and h0.450=0.1538.
.ML.BS.BL.BGaq−112−112α0.297 50.315 90.321 30.310 70.305 80.315 90.283 20.266 9β1.552 11.467 91.507 41.427 91.389 21.467 91.410 81.382 0st0.927 10.918 70.918 90.918 40.918 10.918 70.918 00.917 7ht0.148 60.165 30.166 50.164 10.162 90.165 30.150 40.142 8After the following integration,(56)∫0.21937.994334.52352k∑i=1mRi+1log1+xi:m:n:kR/βdβ.We obtain the confidence area atj=2, by 74.214 1. Similarly, the confidence areas for some values of j are presented in Table 4. Figure 1 shows the 95% confidence region for β and α.Table 4
The interval lengths forβ and 95% confidence area for α and β.
jLengthArea17.3419198.23826.110774.214135.779661.235744.337166.982154.750864.871464.289259.476174.254755.247884.068762.479094.354167.2178105.101755.4785115.189961.2587124.645756.4512135.247542.8979144.562639.4872156.984741.2789Figure 1
Joint confidence region forβ and α.


## 6. Simulation Study

To compare the proposed BEs with the MLEs, a simulation study is performed using various combinations ofn, m, and k and different censored schemes of R (different Ri values). A Pro-F-F-C sample from GPD with the parameters α,β=0.5,2,0.3,1 is generated. The true values of st and ht at time t=0.4 and 0.5 are evaluated to be st=0.9129,ht=0.2083 and st=0.8855,ht=0.2. The performance of the resulting estimators of α, β, st, and ht has been considered in terms of the mean squared error (MSE), which are computed, for l=1,2,3,4, M=1000, ϕ1=α, ϕ2=β, ϕ3=st, and ϕ4=ht as MSE=1/M∑j=1Mϕ^lj−ϕl2. These results were obtained using Mathematica ver. 13. Considering two different group sizes k=1,5 and the following censoring schemes,Scheme I:R1=n−mandRi=0 for i≠1Scheme II:Rm+1/2=n−mandRi=0 for i≠m+1/2 if m odd, and Rm/2=n−mandRi=0 for i≠m/2 if m evenScheme III:Rm=n−mandRi=0 for i≠mThe results of MSE of estimates are reported in Tables5 and 6.Table 5
MSE of MLEs and BEs with true values.
knmC.SMLBSBLBGaq−11−1113020Iα˜0.157 60.092 00.092 10.092 00.092 00.091 9β˜0.788 40.571 10.574 40.566 90.571 10.565 5s˜t0.025 60.022 30.022 90.022 30.022 30.021 6h˜t0.005 60.004 20.004 20.004 10.004 20.004 0IIα˜0.169 90.092 20.092 20.092 10.092 20.091 9β˜0.826 80.573 50.577 00.568 00.573 50.558 4s˜t0.025 80.023 60.023 60.023 10.023 60.022 1h˜t0.005 80.004 40.004 40.004 20.004 40.004 1IIIα˜0.180 10.095 60.095 70.093 90.095 60.093 8β˜0.831 40.578 80.578 90.569 50.578 80.559 5s˜t0.025 90.023 70.023 90.023 50.023 70.022 7h˜t0.006 10.004 60.004 80.004 50.004 60.004 353020Iα˜0.158 80.095 10.095 20.094 80.095 10.094 4β˜0.645 30.571 70.574 80.567 30.571 70.566 3s˜t0.016 00.015 10.014 90.014 70.015 10.014 6h˜t0.004 90.004 10.004 20.004 00.004 10.003 9IIα˜0.159 70.096 70.096 50.096 30.096 70.095 3β˜0.668 00.572 40.575 10.568 20.572 40.567 4s˜t0.016 80.015 20.015 30.014 90.015 20.014 8h˜t0.005 20.004 60.004 90.004 50.004 60.004 3IIIα˜0.160 20.096 90.097 60.096 80.096 90.095 7β˜0.669 20.575 50.578 30.568 40.575 50.568 3s˜t0.017 50.015 70.015 80.015 50.015 70.015 2h˜t0.005 70.004 70.005 10.004 80.004 70.004 513025Iα˜0.150 10.088 60.088 90.088 50.088 60.088 1β˜0.786 10.567 90.568 00.559 30.567 90.527 5s˜t0.025 40.021 80.022 20.021 70.021 80.021 5h˜t0.005 30.004 00.004 10.003 90.004 00.003 6IIα˜0.152 40.089 70.089 80.088 90.089 70.088 6β˜0.795 20.568 30.568 50.559 60.568 30.529 7s˜t0.025 90.023 10.023 90.022 90.023 10.022 7h˜t0.005 40.004 10.004 20.004 00.004 10.003 8IIIα˜0.153 70.089 80.089 90.089 20.089 80.088 9β˜0.795 90.569 20.569 50.560 80.569 20.534 3s˜t0.026 60.024 60.024 80.024 50.024 60.023 9h˜t0.005 50.004 50.004 80.004 40.004 50.004 153025Iα˜0.150 40.089 40.089 80.089 30.089 40.088 3β˜0.642 20.568 20.568 70.559 50.568 20.536 1s˜t0.015 20.014 30.014 40.014 20.014 30.014 1h˜t0.004 10.003 80.003 90.003 70.003 80.003 4IIα˜0.151 70.092 60.092 80.092 50.092 60.092 3β˜0.652 10.569 10.569 50.559 80.569 10.536 9s˜t0.016 10.014 60.014 50.014 30.014 60.014 2h˜t0.004 70.004 10.004 20.003 90.004 10.003 7IIIα˜0.153 70.095 40.095 90.095 10.095 40.093 6β˜0.654 70.569 50.570 70.561 70.569 50.548 7s˜t0.016 60.014 90.015 20.015 10.014 90.014 3h˜t0.005 20.004 50.004 70.004 20.004 50.003 9Table 6
MSE of MLEs and BEs with true values.
knmC.SMLBSBLBGaq−22−2215030Iα˜0.091 40.079 40.083 40.075 60.083 80.067 4β˜0.653 80.413 00.487 10.342 50.438 90.335 2s˜t0.024 20.022 20.022 20.022 20.022 20.022 2h˜t0.005 40.003 80.003 70.003 80.003 70.003 6IIα˜0.094 20.079 80.087 90.078 80.088 30.068 6β˜0.654 10.414 00.487 80.345 00.439 70.337 9s˜t0.025 50.023 10.023 10.023 10.023 10.023 1h˜t0.005 80.003 90.004 00.003 80.004 10.003 7IIIα˜0.094 50.080 60.088 80.079 60.089 10.070 8β˜0.655 80.430 40.505 20.357 50.456 60.350 3s˜t0.027 90.024 50.024 50.024 50.024 50.024 5h˜t0.005 90.004 10.004 20.004 00.004 30.003 955030Iα˜0.091 90.081 20.084 30.077 10.084 70.069 8β˜0.648 30.417 50.487 90.363 30.463 90.356 0s˜t0.019 40.018 20.018 20.018 20.018 20.018 2h˜t0.005 30.003 40.003 50.003 40.003 50.003 4IIα˜0.092 30.082 60.086 10.079 70.088 70.072 7β˜0.649 90.426 30.491 70.372 10.492 70.364 8s˜t0.019 60.018 40.018 40.018 40.018 40.018 4h˜t0.005 40.003 70.003 80.003 70.003 80.003 5IIIα˜0.094 10.084 30.087 70.080 60.089 10.075 6β˜0.651 20.434 40.504 20.385 20.498 80.377 8s˜t0.019 80.018 70.018 70.018 80.018 70.018 8h˜t0.005 50.003 80.004 00.003 90.004 00.003 7


## 7. Conclusion

The main aim of this article is to develop different methods to estimate the unknown quantities of the GPD based on a Pro-F-F-C scheme, which was introduced by Wu and Kuş [30]. We applied the classical and the Bayesian inferential procedures for the unknown parameters and reliability measures. The ACIs have been derived based on the asymptotic normality of MLEs. Under the Bayesian approach, we obtained the BEs based on the SE, LINEX, and GE loss functions. Furthermore, we assumed the conjugate gamma prior for the shape parameter and discrete prior for the scale parameter. The exact confidence interval and exact confidence region for the estimators have been constructed based on pivotal quantities. A numerical example using a simulated data set has been studied to show the practicality of these proposed procedures. The performance of the different estimation methods is realized via a simulation study which is revealed in the following:(1)
The BEs based on SE, LINEX, and GE loss functions perform better than the MLEs, in terms of MSEs(2)
The BEs based on LINEX and GE loss functions whena=1 and 2 and q=1 and 2 perform better than BEs based on SE, in terms of MSEs(3)
The BEs based on the SE loss function perform better than BEs based on LINEX and GE loss functions whena=−1 and −2 and q=−1 and −2, in terms of MSEs(4)
From Tables5 and 6, for a fixed scheme, the MSE values of all estimates, a model’s parameters, and the reliability measures decrease as m/n increases which is consistent with the statistical theory that the larger the sample size, the more accurate the estimate(5)
It can be seen from Tables5 and 6 that the three CS methods vary in terms of preference and sometimes CS I is the best while at other times the CS II or III is the best in the sense of having smaller MSEs(6)
The MSEs forα and β estimates based on the Pro-F-F-C scheme with k=5 increase in those for P-type-II-C with k=1 while the MSEs for st and ht estimates based on the Pro-F-F-C scheme with k=5 decrease in those for P-type-II-C with k=1



---

*Source: 9325928-2021-12-07.xml*