# Estimations of Generalized Exponential Distribution Parameters Based on Type I Generalized Progressive Hybrid Censored Data

Authors: M. Nagy; Adel Fahad Alrasheedi
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

Type I generalized progressive hybrid censoring scheme is a combination of Type I and Type II progressive hybrid censoring schemes, and it is one of the most recent advancements in data censoring. In this article, based on Type I generalized progressive hybrid censoring data from generalized exponential distribution, the maximum likelihood and Bayesian estimators of distribution’s parameters as well as the reliability and hazard functions are approximately calculated. Also, the credible interval estimators of these quantities are obtained. Since these quantities cannot be obtained in closed form, so simulation and analysis using a Monte Carlo simulation study with Gibbs sampling are taken. Finally, an illustrative example using real data set is presented to compare the proposed procedures presented and developed here.


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## Body

## 1. Introduction

Computational methods and data processing are among the most important sciences that have different applications in many fields such as medicine, engineering, agriculture, and various vital fields. For a large data, it was necessary to find method for statistical inference by using small samples taken in a certain way of this large data, in order to limit the time and the cost; hence, the idea of censored samples started. There are a variety of scenarios where observed data is censored in nature, including reliability and life testing experiments. The two most popular censoring techniques are Type I and Type II censoring schemes. Epstein [1] initially proposed the hybrid censoring scheme, which is a mixture of Type I and Type II censoring schemes. One of the main problems of Type I, Type II, and hybrid censoring (HC) methods is that they do not allow for unit removal at whatever point other than the experiment’s finish. To address this issue, a more comprehensive censoring scheme known as progressive Type II censoring was implemented. Balakrishnan and Cramer [2] provide a full discussion of these censoring schemes as well as some recent advances. The disadvantages of the progressive Type II censoring scheme are that the time of the experiment can be very long if the units are highly reliable. Therefore, Kundu and Joarder [3] proposed a progressive hybrid censoring scheme (PHCS). Under progressive hybrid censoring scheme, the time on experiment will be no more than T. Some recent studies on progressive hybrid censoring have been carried out by many authors including Lin et al. [4], Panahi [5], Mohie El-Din et al. [6], and El-Din et al. [7]. On the other side, one of the progressive hybrid censoring scheme’s drawbacks is that it cannot be used when there are only a few failures before time T.Cho et al. [8] suggest a Type I generalized PHCS that allows us to notice a prespecified number of failures for this reason. As a result, a specified number of failures and their lifetimes are always supplied under the Type I generalized PHCS. The life testing experiment based on this censoring scheme can reduce total test duration as well as the cost incurred due to unit failures. Furthermore, because there are more failed observations, statistical estimating efficiency improves. The Type I generalized PHCS have been investigated for instance by El-Din et al. [9], Nagy et al. [10, 11], and Nagy and Alrasheedi [12].We look at statistical inference for a generalized exponential (GE) distribution under Type I generalized PHCS in this paper. Gupta and Kundu [13] provide the probability density function (PDF) and cumulative distribution function (CDF) of the GE distribution, respectively.
(1)fx;α,λ=αλexp−λx1−exp−λxα−1,x,α,λ>0,(2)Fx;α,λ=1−exp−λxα,x,α,λ>0,whereα and λ denote the shape and scale parameters, respectively. For simplicity, let ψx;λ=1−exp−λx; then, PDF and CDF of the GE distribution can be rewritten, respectively, by the following:
(3)fx;α,λ=αλexp−λxψx;λα−1,x,α,λ>0,(4)Fx;α,λ=ψx;λα,x,α,λ>0.Therefore, the reliability (survival) and hazard functions may be written, respectively, as follows:(5)Sx;α,λ=1−ψx;λα,x,α,λ>0,(6)Hx;α,λ=fx;α,λSx;α,λ=αλexp−λxψx;λα−11−ψx;λα,x,α,λ>0.The GE distribution is a very flexible and favorably skewed model that is used to replace the lognormal, gamma, and Weibull distributions. It is often used to examine data that is skewed in a positive direction. Inferences for the GE distribution have been discussed by many authors including Gupta and Kundu [14] and Jaheen [15].The aim of this article is that we consider the analysis of the Type I progressive censoring data from generalized exponential distribution to calculate the maximum likelihood and Bayesian estimators of unknown parameters and also calculate the approximate confidence intervals. The remainder of this paper is as follows. In Section2, we provide an overview of the Type I generalized PHCS and compute the likelihood function of the model based on Type I generalized PHCS. The ML estimators with the corresponding approximate confidence intervals of the GE distribution’s parameters, the reliability function and hazard function, are derived in Section 3. In Section 4, we use MCMC with the Gibbs sampling procedure to compute the Bayesian estimates of the parameters, reliability and hazard functions for the GE distribution, and also construct their credible intervals. Under Type I generalized censoring scheme, simulation studies to compare the efficacy of the offered inference methodologies are carried out in Section 5. In Section 6, some computational results with real data are presented for illustrating all the inferential methods developed here. Finally, Section 7 concludes the paper.


## 2. The Likelihood Model Description

Consider life testing experiment in whichn equivalent units are tested. For T∈0,∞ and integers k,m∈1,2,⋯,n and R=R1,⋯−Rm are prefixed such thatk<m. Let d denote the number of observed failures up to time T, and Yk:m:n and Ym:m:n are the times of failurek,m, respectively. According to the Type I generalized PHCS which introduced by Cho et al. [8], the experiment termination time is T∗=maxYk:m:n,minYm:m:n,T and we have the following observations:
(1)
SupposeT is reached before the kthfailure, then the experiment terminates at Yk:m:n and we will observe Y1:m:n<⋯<Yd:m:n<Yd+1:m:n<⋯<Yk:m:n(2)
Suppose that theT is reached between the kth and mth, then the experiment terminates at T and we will observe Y1:m:n<⋯<Yk:m:n<Yk+1:m:n<⋯<YD:m:n(3)
Suppose that themthfailure occurs before Ti.e.,Ym:m:n≤T, then the experiment terminates at Xm:m:n and we will observe Y1:m:n<⋯<Yk:m:n<Yk+1:m:n<⋯<Ym:m:nThus, the joint density function based on the above cases can be written as follows:(7)fY¯=Ci,j∏i=1D∗fyi:m:nSyi:m:nRi∗STRτ∗,where
(8)Ci,j=∏i=1D∗∑j=1mRj∗,D∗=maxk,minm,d,R∗=R1,⋯,Rd,⋯,Rk,ifT<Yk:m:n,R1,⋯,Rk,⋯,Rd,ifYk:m:n<T<Ym:m:n,R1,⋯,Rk,⋯,Rm,ifYk:m:n<T,where Rτ∗ is the number of surviving units eliminated at T, as determined by the following:
(9)Rτ∗=0,ifT<Yk:m:n,∑j=1dRj,ifYk:m:n<T<Ym:m:n,0,ifYk:m:n<T,with
(10)y¯=y1,⋯,yd,⋯,yk,ifT<Yk:m:n,y1,⋯,yk,⋯,yd,ifYk:m:n<T<Ym:m:n,y1,⋯,yk,⋯,ym,ifYk:m:n<T,andxi=xi:m:n for simplicity of notation.Upon using Equations (3) and (4) in Equation (7), the likelihood function of α,β based on generalized Type I HPCS can be obtained as follows:
(11)Lα,λ∣y¯=Ci,jαD∗λD∗exp−λ∑i=1D∗yiexpα−1∑i=1D∗lnψyi;λ×expRτ∗ln1−ψT;λα+∑i=1D∗Ri∗ln1−ψyi;λα.


## 3. The ML Estimations

For computing the ML estimates, we must maximize the likelihoodLα,λ∣y¯ with respect to α and λ in order to compute the ML estimates. The logarithm of the likelihood function is given by:
(12)ℓα,λ∣y¯∝D∗lnα+D∗lnλ−λ∑i=1D∗yi+α−1∑i=1D∗lnψyi;λ+∑i=1D∗Ri∗ln1−ψyi;λα+Rτ∗ln1−ψT;λα.The ML estimatesα^ML and λ^ML of α and λ can be obtained by simultaneously solving the following equations.
(13)∂ℓα,λ∣y¯∂α=D∗α+∑i=1D∗lnψyi;λ−∑i=1D∗Ri∗ψyi;λαlnψyi;λ1−ψyi;λα−Rτ∗ψT;λαlnψT;λ1−ψT;λα=0,(14)∂ℓα,λ∣y¯∂λ=D∗λ−∑i=1D∗yi+α−1∑i=1D∗yi1−ψyi;λψyi;λ−α∑i=1D∗Ri∗yi1−ψyi;λψyi;λα1−ψyi;λα−αRτ∗T1−ψT;λψT;λα1−ψT;λα=0.By using the invariance property of the ML estimator, the ML estimators of the corresponding reliability and hazard functions are given, respectively, by(15)S^MLt=1−ψt;λ∧MLα∧ML−1,t>0,(16)H^MLt=α^MLλ^MLexp−λ^MLtψyi;λ∧MLα∧ML−11−ψt;λ∧MLα∧ML.


### 3.1. Approximate Confidence Intervals forα and λ

For large observationD∗, the observed Fisher information matrix of the parameters α and λ is given by the following:
(17)Iα^ML,λ^ML=−∂2ℓα,λ∣y¯∂α2−∂2ℓα,λ∣y¯∂α∂λ−∂2ℓα,λ∣y¯∂α∂λ−∂2ℓα,λ∣y¯∂λ2α=α^ML,λ=λ^ML,where
(18)∂2ℓα,λ∣y¯∂α2=−D∗α2−∑i=1D∗Ri∗ψyi;λ2αlnψyi;λ21−ψyi;λα2+ψyi;λαlnψyi;λ21−ψyi;λα−Rτ∗ψT;λ2αlnψT;λ21−ψT;λα2+ψT;λαlnψT;λ21−ψT;λα,(19)∂2ℓα,λ∣y¯∂α∂λ=∑i=1D∗xi1−ψyi;λψyi;λ−∑i=1D∗Ri∗αyi1−ψyi;λψyi;λαlnψyi;λψyi;λ1−ψyi;λα+xi1−ψyi;λψyi;λαψyi;λ1−ψyi;λα+αyi1−ψyi;λψyi;λ2αlnψyi;λψyi;λ1−ψyi;λα2−Rτ∗αT1−ψT;λψT;λαlnψT;λψT;λ1−ψT;λα+T1−ψT;λψT;λαψT;λ1−ψT;λα+αT1−ψT;λψT;λ2αlnψT;λψT;λ1−ψT;λα2,(20)∂2ℓα,λ∣y¯∂λ2=−mλ2−α−1∑i=1D∗xi21−ψyi;λψyi;λ+xi21−ψyi;λ2ψyi;λ2+∑i=1D∗Ri∗αyi21−ψyi;λψyi;λαψyi;λ1−ψT;λα−α2yi21−ψyi;λψyi;λαψyi;λ21−ψyi;λα+αyi21−ψyi;λ2ψyi;λαψyi;λ21−ψT;λα−α2yi21−ψyi;λ2ψyi;λ2αψyi;λ21−ψyi;λα2+Rτ∗αT21−ψT;λψxi;λαψT;λ1−ψT;λα−α2T21−ψT;λψT;λαψT;λ21−ψT;λα+αT21−ψT;λ2ψT;λαψT;λ21−ψT;λα−α2T21−ψT;λ2ψT;λ2αψT;λ21−ψT;λα2.Based on the normal approximation of the ML estimators,α^−α/Vα^ and λ^−λ/Vλ^ are asymptotically normally distributed with mean 0 and variance 1; that is,
(21)α^−αVα^~N0,1,λ^−λVλ^~N0,1.Two-sided1−γ 100% confidence intervals for α and λ are given by α^±Zγ/2Vα^ and λ^±Zγ/2Vλ^, respectively, where Vα^ and Vλ^ are the asymptotic variances of α^ and λ^, respectively, obtained by inverting the matrix Iα^ML,λ^ML, and Zγ/2 is the upperγth/2 percentile of the standard normal distribution.


### 3.2. Approximate Confidence Intervals forSt and Ht

In this subsection, we calculate the estimated confidence intervals for the reliability and hazard functions using the delta method proposed by Greene [16]. The delta method is a general method for calculating confidence intervals for MLE functions that are too complex to calculate the variance analytically. It creates a linear approximation of the function and then calculates the variance of the simpler linear function that can be used for large sample inference (see [17]). Let
(22)M1=∂St∂α∂St∂λ,M2=∂Ht∂α∂Ht∂λ,where
(23)∂St∂α=−ψt;λαlnψt;λ,(24)∂St∂λ=tλψt;λα1−ψt;λψt;λ,(25)∂Ht∂α=λ1−ψt;λψt;λα−1+αλψt;λα−1lnψt;λ1−ψt;λα+αλ1−ψt;λψt;λα−1lnψt;λ1−ψt;λα2,(26)∂Ht∂λ=α1−ψt;λψt;λα−1−αλt1−ψt;λψt;λα−11−ψt;λα+tλ21−ψt;λ2ψt;λ2α−2ψt;λ1−ψt;λα+λα2t1−ψt;λψt;λ2α−1ψt;λ1−ψt;λα2.Then, the approximate estimates ofVSt^ and VHt^ are given, respectively, by the following:
(27)VSt^≅M1tI−1M1α=α^ML,λ=λ^ML,(28)VHt^≅M2tI−1M2α=α^ML,λ=λ^ML,where M1t and M2t are the transposes of the matrix M1 and M2, respectively. Again, based on the normal approximation of the ML estimators, St^−St/VSt^ and Ht^−Ht/VHt^ are asymptotically normally distributed with mean 0 and variance 1; these results yield 1−γ 100% approximate confidence intervals for St and Ht which are given by the following.
(29)St^±Zγ/2VSt^,Ht^±Zγ/2VHt^.For more on different types of confidence intervals, we refer our readers to Banik et al. [18] and Almonte and Kibria [19] among others.


## 3.1. Approximate Confidence Intervals forα and λ

For large observationD∗, the observed Fisher information matrix of the parameters α and λ is given by the following:
(17)Iα^ML,λ^ML=−∂2ℓα,λ∣y¯∂α2−∂2ℓα,λ∣y¯∂α∂λ−∂2ℓα,λ∣y¯∂α∂λ−∂2ℓα,λ∣y¯∂λ2α=α^ML,λ=λ^ML,where
(18)∂2ℓα,λ∣y¯∂α2=−D∗α2−∑i=1D∗Ri∗ψyi;λ2αlnψyi;λ21−ψyi;λα2+ψyi;λαlnψyi;λ21−ψyi;λα−Rτ∗ψT;λ2αlnψT;λ21−ψT;λα2+ψT;λαlnψT;λ21−ψT;λα,(19)∂2ℓα,λ∣y¯∂α∂λ=∑i=1D∗xi1−ψyi;λψyi;λ−∑i=1D∗Ri∗αyi1−ψyi;λψyi;λαlnψyi;λψyi;λ1−ψyi;λα+xi1−ψyi;λψyi;λαψyi;λ1−ψyi;λα+αyi1−ψyi;λψyi;λ2αlnψyi;λψyi;λ1−ψyi;λα2−Rτ∗αT1−ψT;λψT;λαlnψT;λψT;λ1−ψT;λα+T1−ψT;λψT;λαψT;λ1−ψT;λα+αT1−ψT;λψT;λ2αlnψT;λψT;λ1−ψT;λα2,(20)∂2ℓα,λ∣y¯∂λ2=−mλ2−α−1∑i=1D∗xi21−ψyi;λψyi;λ+xi21−ψyi;λ2ψyi;λ2+∑i=1D∗Ri∗αyi21−ψyi;λψyi;λαψyi;λ1−ψT;λα−α2yi21−ψyi;λψyi;λαψyi;λ21−ψyi;λα+αyi21−ψyi;λ2ψyi;λαψyi;λ21−ψT;λα−α2yi21−ψyi;λ2ψyi;λ2αψyi;λ21−ψyi;λα2+Rτ∗αT21−ψT;λψxi;λαψT;λ1−ψT;λα−α2T21−ψT;λψT;λαψT;λ21−ψT;λα+αT21−ψT;λ2ψT;λαψT;λ21−ψT;λα−α2T21−ψT;λ2ψT;λ2αψT;λ21−ψT;λα2.Based on the normal approximation of the ML estimators,α^−α/Vα^ and λ^−λ/Vλ^ are asymptotically normally distributed with mean 0 and variance 1; that is,
(21)α^−αVα^~N0,1,λ^−λVλ^~N0,1.Two-sided1−γ 100% confidence intervals for α and λ are given by α^±Zγ/2Vα^ and λ^±Zγ/2Vλ^, respectively, where Vα^ and Vλ^ are the asymptotic variances of α^ and λ^, respectively, obtained by inverting the matrix Iα^ML,λ^ML, and Zγ/2 is the upperγth/2 percentile of the standard normal distribution.


## 3.2. Approximate Confidence Intervals forSt and Ht

In this subsection, we calculate the estimated confidence intervals for the reliability and hazard functions using the delta method proposed by Greene [16]. The delta method is a general method for calculating confidence intervals for MLE functions that are too complex to calculate the variance analytically. It creates a linear approximation of the function and then calculates the variance of the simpler linear function that can be used for large sample inference (see [17]). Let
(22)M1=∂St∂α∂St∂λ,M2=∂Ht∂α∂Ht∂λ,where
(23)∂St∂α=−ψt;λαlnψt;λ,(24)∂St∂λ=tλψt;λα1−ψt;λψt;λ,(25)∂Ht∂α=λ1−ψt;λψt;λα−1+αλψt;λα−1lnψt;λ1−ψt;λα+αλ1−ψt;λψt;λα−1lnψt;λ1−ψt;λα2,(26)∂Ht∂λ=α1−ψt;λψt;λα−1−αλt1−ψt;λψt;λα−11−ψt;λα+tλ21−ψt;λ2ψt;λ2α−2ψt;λ1−ψt;λα+λα2t1−ψt;λψt;λ2α−1ψt;λ1−ψt;λα2.Then, the approximate estimates ofVSt^ and VHt^ are given, respectively, by the following:
(27)VSt^≅M1tI−1M1α=α^ML,λ=λ^ML,(28)VHt^≅M2tI−1M2α=α^ML,λ=λ^ML,where M1t and M2t are the transposes of the matrix M1 and M2, respectively. Again, based on the normal approximation of the ML estimators, St^−St/VSt^ and Ht^−Ht/VHt^ are asymptotically normally distributed with mean 0 and variance 1; these results yield 1−γ 100% approximate confidence intervals for St and Ht which are given by the following.
(29)St^±Zγ/2VSt^,Ht^±Zγ/2VHt^.For more on different types of confidence intervals, we refer our readers to Banik et al. [18] and Almonte and Kibria [19] among others.


## 4. Bayesian Estimates

In this section, we derive the Bayesian estimates of the GE distribution’s parametersα andλ, based on Type I generalized HPCS data. For the Bayesian estimations, it is under the premise that both distribution’s parameters α and λ are independent and have gamma prior distributions, respectively,
(30)πα=b1a1Γa1αa1−1exp−αb1,πλ=b2a2Γa2λa2−1exp−λb2,where hyper parameters ai,bi for all i=1,2 are the positive real constants that indicate prior knowledge of distribution’s parameters; if ai,bi for all i=1,2 are set to be equal zero, then the informative priors πα and πλ are reduced to the noninformative priors. The joint prior density of α and λ can be obtained by multiplying πα by πλ as follows:
(31)πα,λ=b1a1b2a2Γa1Γa2αa1−1λa2−1exp−αb1−λb2.Upon combining Equations (11) and (31), given the Type I generalized PHCS, the corresponding posterior distribution of α,λ is obtained as follows:
(32)π∗α,λ∣y¯=Lα,λ∣y¯πα,λ∫0x0∫0∞Lα,λ∣y¯πα,λdαdλ∝αD∗+a1−1λD∗+a2−1exp−λ∑i=1D∗yiexpα−1∑i=1D∗lnψyi;λ×exp−αb1−λb2expRτ∗ln1−ψT;λα+∑i=1D∗Ri∗ln1−ψyi;λα.The Bayesian estimate of a parametric functiongθ under squared error loss (SEL) function is obtained as follows:
(33)gθ^BS=∫0∞∫0∞gθπ∗α,β∣y¯dαdλ.All Bayesian estimators based on the SEL function are in the form of a ratio of two integrals for which there are no closed-form solutions, so to estimate the above integrals, we must use appropriate numerical methods. To compute Bayesian estimates and build credible intervals for the distribution’s parametersα andλ, we apply here the MCMC approaches. When compared to previous methods, the MCMC method gives an alternate way for parameter estimate that is more flexible. For more details about the MCMC approaches, for further details, see [20, 21].


### 4.1. The Metropolis-Hastings Algorithm within Gibbs Sampling

The Metropolis-Hastings (MH) algorithm was presented by Metropolis et al. [22] as a general Markov chain Monte Carlo (MCMC) approach, and Hastings [23] developed the MH algorithm. The MH approach can be used to generate random samples from any arbitrary complicated target distribution with any dimension that is known up to a normalizing constant. The Gibbs sampling method is a subset of the MCMC method. It can be used to produce a sequence of samples from two or more random variables’ entire conditional probability distributions. Decomposing the joint posterior distribution into entire conditional distributions for each parameter and sample from them is required for Gibbs sampling. To produce a sample from the posterior density functionπ∗α,λ∣y¯, we propose utilizing the Gibbs sampling approach.From Equation (32), the conditional posterior density function of α givenπ∗α∣λ,y¯ can be obtained as follows:
(34)π∗α∣λ,y¯∝αD∗+a1−1expα−1∑i=1D∗lnψyi;λ×expRτ∗ln1−ψT;λα+∑i=1D∗Ri∗ln1−ψyi;λα.Similarly, the conditional posterior density function ofλ given π∗α∣λ,y¯ can be obtained as follows:
(35)π∗λ∣α,y¯∝λD∗+a2−1exp−λb2exp−λ∑i=1D∗yiexpα−1∑i=1D∗lnψyi;λ×expRτ∗ln1−ψT;λα+∑i=1D∗Ri∗ln1−ψyi;λα.Furthermore, the conditional posterior distributions ofα and λ in Equations (34) and (35) cannot be reduced analytically to well-known distributions; therefore, they cannot be sampled directly using usual methods, but their plots show that they are similar to normal distributions. We employ the MH technique within the Gibbs sampling scheme with normal proposal distribution to generate random numbers from these distributions. Now, we propose the following scheme for generating and deriving the posterior density functions, as well as the Bayesian estimates and credible intervals.
(1)
Step 1: Start withα,λ as α0,λ0=α^ML,λ^ML(2)
Step 2: Fori=1,2,⋯,N, repeat the following steps(i)
Setα,λ=αi−1,λi−1(ii)
Generateαi from π∗α∣λ,y¯ given in Equation (33) with the Nαi−1,Vα^ proposal distribution(iii)
Generateλi from π∗λ∣α,y¯ given in Equation (35) with the Nλi−1,Vλ^ proposal distribution(iv)
Calculate the probabilitiespα=min1,π∗α∗∣λ∗,y¯/π∗α∣λ,y¯(v)
Calculate the probabilitiespλ=min1,π∗λ∗∣α∗,y¯/π∗λ∣α,y¯(vi)
Setαi=α∗ with probability pα; otherwise, set αi=α(vii)
Setλi=λ∗ with probability pλ; otherwise, set λi=λ(viii)
Seti=i+1Finally, some initial observations of sizeB, say, are eliminated as burn-in observations from a total N generated observations of α and λ, and the remaining N‐B observations can be used to compute Bayesian estimates. The required Bayesian estimates of α under SELF are computed as follows:
(36)α^BS=1N−B∑i=B+1N−Bαi.The Bayesian estimate of λ under SELF is given by the following:
(37)α^BS=1N−B∑i=B+1N−Bλi.Substitutingαi and λi into Equations (5) and (6) to compute Sit and Hit, for i=1,2,⋯,N, then the Bayesian estimates of St and Ht under SELF are computed, respectively, as follows:
(38)St^BS=1N−B∑i=B+1N−BSii,(39)Ht^BS=1N−B∑i=B+1N−BHii.This algorithm is also useful in computing credible intervals of unknown parameters.1−γ 100% symmetric credible intervals of α, λ, St, and Ht become, respectively, the following:
(40)αN−Bγ/2,αN−B1−γ/2,(41)λN−Bγ/2,λN−B1−γ/2,(42)SN−Bγ/2t,SN−B1−γ/2t,(43)HN−Bγ/2t,HN−B1−γ/2t.


## 4.1. The Metropolis-Hastings Algorithm within Gibbs Sampling

The Metropolis-Hastings (MH) algorithm was presented by Metropolis et al. [22] as a general Markov chain Monte Carlo (MCMC) approach, and Hastings [23] developed the MH algorithm. The MH approach can be used to generate random samples from any arbitrary complicated target distribution with any dimension that is known up to a normalizing constant. The Gibbs sampling method is a subset of the MCMC method. It can be used to produce a sequence of samples from two or more random variables’ entire conditional probability distributions. Decomposing the joint posterior distribution into entire conditional distributions for each parameter and sample from them is required for Gibbs sampling. To produce a sample from the posterior density functionπ∗α,λ∣y¯, we propose utilizing the Gibbs sampling approach.From Equation (32), the conditional posterior density function of α givenπ∗α∣λ,y¯ can be obtained as follows:
(34)π∗α∣λ,y¯∝αD∗+a1−1expα−1∑i=1D∗lnψyi;λ×expRτ∗ln1−ψT;λα+∑i=1D∗Ri∗ln1−ψyi;λα.Similarly, the conditional posterior density function ofλ given π∗α∣λ,y¯ can be obtained as follows:
(35)π∗λ∣α,y¯∝λD∗+a2−1exp−λb2exp−λ∑i=1D∗yiexpα−1∑i=1D∗lnψyi;λ×expRτ∗ln1−ψT;λα+∑i=1D∗Ri∗ln1−ψyi;λα.Furthermore, the conditional posterior distributions ofα and λ in Equations (34) and (35) cannot be reduced analytically to well-known distributions; therefore, they cannot be sampled directly using usual methods, but their plots show that they are similar to normal distributions. We employ the MH technique within the Gibbs sampling scheme with normal proposal distribution to generate random numbers from these distributions. Now, we propose the following scheme for generating and deriving the posterior density functions, as well as the Bayesian estimates and credible intervals.
(1)
Step 1: Start withα,λ as α0,λ0=α^ML,λ^ML(2)
Step 2: Fori=1,2,⋯,N, repeat the following steps(i)
Setα,λ=αi−1,λi−1(ii)
Generateαi from π∗α∣λ,y¯ given in Equation (33) with the Nαi−1,Vα^ proposal distribution(iii)
Generateλi from π∗λ∣α,y¯ given in Equation (35) with the Nλi−1,Vλ^ proposal distribution(iv)
Calculate the probabilitiespα=min1,π∗α∗∣λ∗,y¯/π∗α∣λ,y¯(v)
Calculate the probabilitiespλ=min1,π∗λ∗∣α∗,y¯/π∗λ∣α,y¯(vi)
Setαi=α∗ with probability pα; otherwise, set αi=α(vii)
Setλi=λ∗ with probability pλ; otherwise, set λi=λ(viii)
Seti=i+1Finally, some initial observations of sizeB, say, are eliminated as burn-in observations from a total N generated observations of α and λ, and the remaining N‐B observations can be used to compute Bayesian estimates. The required Bayesian estimates of α under SELF are computed as follows:
(36)α^BS=1N−B∑i=B+1N−Bαi.The Bayesian estimate of λ under SELF is given by the following:
(37)α^BS=1N−B∑i=B+1N−Bλi.Substitutingαi and λi into Equations (5) and (6) to compute Sit and Hit, for i=1,2,⋯,N, then the Bayesian estimates of St and Ht under SELF are computed, respectively, as follows:
(38)St^BS=1N−B∑i=B+1N−BSii,(39)Ht^BS=1N−B∑i=B+1N−BHii.This algorithm is also useful in computing credible intervals of unknown parameters.1−γ 100% symmetric credible intervals of α, λ, St, and Ht become, respectively, the following:
(40)αN−Bγ/2,αN−B1−γ/2,(41)λN−Bγ/2,λN−B1−γ/2,(42)SN−Bγ/2t,SN−B1−γ/2t,(43)HN−Bγ/2t,HN−B1−γ/2t.


## 5. Simulation Study

In this section, to evaluate the performance of the proposed methods of different estimators of the parameters presented in the preceding sections, a simulation study is carried out. Without losing generality, sample of size (n=60) is used for the simulation study with different sample sizes (k,m, wherem=2k) and different values of Tand the following censoring schemes:
(i)
Scheme 1:R1=Rk=2n−m/5,Rm=n−m/5.(ii)
Scheme 2:R1=Rm=2n−m/5,Rk=n−m/5.(iii)
Scheme 1:Rk=Rm=2n−m/5,R1=n−m/5.We then generate Type I generalized PHCSs from a GE distribution withα=2 and λ=1.2. To compute the Bayesian estimates and the credible intervals using MCMC algorithm,by using the number of iterations N=11,000, the ML estimates for unknown parameters α and λ have been used as initial values for running the MCMC algorithm. The first values of the generated sequences may be far from reminded converged sequences, so the first B=1,000 values are removed here to avoid the effects of the initial values and the procedure repeated 1000 times. We use the informative gamma priors for the two distribution’s parameters α, λ with hyper parameters a1=a2=1 and b1=b2=2 (prior 1) and noninformative priors with a1=b1=a2=b2=0 (prior 2). Performance of different point estimates is compared in terms of their mean squared error (MSE) and estimate bias (EB) values for α, λ, St, and Ht with t=0.6, as shown in Tables 1–4, respectively. Also, the average length (AL) and coverage probability (CP) of the 95% approximate confidence intervals are displayed in Tables 5–8.Table 1
The mean and MSE values of the ML and Bayesian estimates ofα.
MLBayesianPrior 1Prior 2k,m,nTSchemeMeanMSEMeanMSEMeanMSE15,30,600.5012.12550.48602.06060.32391.99080.352920,40,602.15010.43152.12990.29592.03650.384225,50,602.16360.28152.14710.24482.04990.260715,30,6022.18630.38592.11920.25791.98170.273320,40,602.22490.50322.22170.24712.06460.356225,50,602.04890.23322.04760.18981.94530.198915,30,6032.13560.36122.08140.32261.98160.333920,40,602.23640.43842.23130.38722.07410.409125,50,602.14300.25122.14910.18082.00830.186715,30,601.0012.16000.42852.10120.32072.07060.349820,40,602.20210.34542.24080.21042.04260.278025,50,602.08010.26212.03320.20651.95710.215215,30,6022.24590.62462.28900.38732.11730.422720,40,602.17060.41802.18670.29282.06390.293525,50,602.04080.28172.08780.15111.95690.247015,30,6032.06410.41762.05290.39361.92820.402220,40,602.15150.33302.13660.24582.02150.310825,50,602.21390.32062.17950.18512.07840.229015,30,601.5012.09890.29122.02750.23471.93010.262820,40,602.14480.19872.09420.15031.97120.154425,50,601.99330.19742.05140.10421.95060.172415,30,6022.17190.65352.19960.37901.97040.389920,40,602.19070.41952.23180.24462.07460.309425,50,602.18200.38452.15590.23381.99890.278915,30,6032.17720.42012.13990.20941.92300.284020,40,602.11740.33132.10830.18201.98680.186925,50,602.11320.23972.07250.19902.02770.2099Table 2
The mean and MSE values of the ML and Bayesian estimates ofλ.
MLBayesianPrior 1Prior 2k,m,nTSchemeMeanMSEMeanMSEMeanMSE15,30,600.5011.26440.21041.39130.12021.37190.184620,40,601.18820.08421.25380.05661.28530.071925,50,601.19520.05191.23350.03281.27260.035815,30,6021.19880.15821.32270.06981.34460.105020,40,601.16580.10671.24460.05541.26160.072025,50,601.22180.07131.26640.04361.29380.063215,30,6031.22770.15581.33460.09991.36280.151320,40,601.18410.09291.23050.0581.29650.087225,50,601.19780.06221.23020.03871.28080.047515,30,601.0011.22100.12411.32100.07361.30240.100520,40,601.14830.05041.17890.03371.22700.03725,50,601.22680.06641.27770.04071.30140.051315,30,6021.13480.07841.18340.05221.23690.070420,40,601.19220.06891.23730.04291.27100.066225,50,601.21430.07691.23330.03981.28950.048815,30,6031.23930.13231.30690.09191.34050.118920,40,601.16920.05461.21820.04361.25800.054125,50,601.16110.04651.21430.0311.22580.036215,30,601.5011.20940.13471.30980.08671.32120.102820,40,601.14980.05051.21080.03711.24470.043325,50,601.24180.04631.26720.02691.29700.041115,30,6021.21220.12281.28620.07211.36100.110820,40,601.15590.05471.18950.03661.24020.051325,50,601.15830.0691.21700.04141.23240.045415,30,6031.15810.09291.25660.05121.31920.086520,40,601.19330.07961.25800.04181.29080.063825,50,601.19200.04861.23770.04111.24550.0470Table 3
The mean and MSE values of the ML and Bayesian estimates ofSt=0.6.
MLBayesianPrior 1Prior 2k,m,nTSchemeMeanMSEMeanMSEMeanMSE15,30,600.5010.85020.04020.85580.03900.85010.039420,40,600.84600.04040.85200.03740.84220.040225,50,600.85390.03740.85730.03420.85000.036715,30,6020.85240.03770.85810.03550.84980.036120,40,600.84950.03950.85550.03390.84800.038825,50,600.84240.03780.84740.03570.84030.037115,30,6030.84690.03430.85540.03270.84740.034020,40,600.85450.03530.85910.03470.85160.035125,50,600.85260.03810.85680.03320.84750.034915,30,601.0010.85140.03610.85600.03090.85200.032820,40,600.84870.03760.85610.0360.84300.036825,50,600.84860.04080.84910.03840.84260.039115,30,6020.84390.05070.85440.04730.84470.049920,40,600.85150.03670.85640.03560.84950.035925,50,600.84150.04680.84590.03990.83750.042515,30,6030.83540.04840.84370.04650.83470.046720,40,600.84540.04090.85010.03580.84050.038825,50,600.85430.03940.85550.03760.84860.038215,30,601.5010.83920.04480.84510.04150.83700.042620,40,600.84380.03760.84580.03480.83710.036425,50,600.84370.03890.85220.03550.84360.038515,30,6020.84950.04270.85840.03860.84590.042620,40,600.84860.03950.85450.03880.84600.03925,50,600.84640.04030.84850.03710.83650.038515,30,6030.84440.03890.85090.03780.83920.038120,40,600.84720.04070.85220.03640.84440.036725,50,600.84670.03620.84840.03240.84360.0354Table 4
The mean and MSE values of the ML and Bayesian estimates ofHt=0.6.
MLBayesianPrior 1Prior 2k,m,nTSchemeMeanMSEMeanMSEMeanMSE15,30,600.5010.45720.10800.42600.09970.43710.104120,40,600.47480.08170.44820.08060.46230.081625,50,600.45410.07590.43830.07000.44610.071215,30,6020.46370.09620.43190.08760.43920.089620,40,600.47340.09370.44860.08510.45860.092825,50,600.47180.07460.45480.07040.46150.071415,30,6030.46890.08360.43460.0790.44260.0820,40,600.45950.07970.44310.07640.44330.077625,50,600.45810.08150.44220.07310.45230.078615,30,601.0010.46270.09120.43470.08190.44600.085420,40,600.47580.07620.45520.07360.46900.073725,50,600.45770.07920.44590.07650.45450.076715,30,6020.48880.11400.45740.10060.46690.113020,40,600.46070.07160.44280.06930.44960.069625,50,600.47630.09150.46090.08290.46860.086715,30,6030.48600.10510.45640.0990.46760.100520,40,600.48020.08580.45830.08140.46810.085625,50,600.46180.08760.44780.08310.45870.085815,30,601.5010.49110.12160.45920.11040.46890.112220,40,600.48880.09670.47230.08350.47910.094325,50,600.46510.08120.44250.07650.45310.077615,30,6020.46400.09900.43060.08920.43960.092120,40,600.47370.08540.45540.07450.46110.081125,50,600.48010.08810.46350.0840.48160.08815,30,6030.48790.10150.45550.08960.46420.093820,40,600.47130.09260.44660.08110.45810.087725,50,600.47200.08070.45760.07870.46510.0796Table 5
The values of AL and corresponding CP of 95% confidence interval for the ML and Bayesian estimates ofα.
MLBayesianPrior 1Prior 2k,m,nTSchemeALCPALCPALCP15,30,600.5012.15030.761.24850.941.29970.8020,40,601.94080.681.10510.981.21340.7025,50,601.79220.721.09890.981.14670.7215,30,6022.21020.721.22330.961.34010.7420,40,602.02270.581.16140.981.16470.7625,50,601.67970.701.04470.941.10230.7415,30,6032.15580.721.24660.941.33930.7820,40,602.04010.521.12210.981.23420.7625,50,601.77300.781.09670.981.10480.7815,30,601.0012.08430.761.26370.921.28850.8220,40,601.94420.701.12190.981.18010.7625,50,601.69100.621.00460.941.10090.8415,30,6022.22030.721.22740.941.37340.7820,40,601.92870.641.11610.961.15250.7825,50,601.65620.701.02830.961.05040.7415,30,6032.00340.681.16860.841.27090.6820,40,601.89920.741.10260.981.17460.7625,50,601.82190.761.09090.981.13810.7815,30,601.5012.01360.741.22140.941.26810.8420,40,601.87810.801.11190.981.14370.8425,50,601.59590.761.04560.981.10860.8015,30,6022.13740.721.23790.941.41950.7820,40,601.94620.601.14320.961.26500.8425,50,601.79370.601.03370.941.05200.8015,30,6032.12350.801.23940.981.33760.8220,40,601.85770.741.17980.961.28030.8225,50,601.71610.781.03690.921.05170.80Table 6
The values of AL and corresponding CP of 95% confidence interval for the ML and Bayesian estimates ofλ.
MLBayesianPrior 1Prior 2k,m,nTSchemeALCPALCPALCP15,30,600.5011.18770.800.99990.941.06300.8020,40,600.89050.780.77660.920.86240.8025,50,600.75860.860.66830.940.71080.9015,30,6021.09730.800.98080.901.03670.8220,40,600.86560.780.78790.900.79860.8225,50,600.78670.840.71460.980.73860.8615,30,6031.14790.800.99620.921.09160.8420,40,600.87990.680.71720.900.83680.8225,50,600.76190.860.65030.960.72240.9015,30,601.0011.02490.780.93540.880.95150.8220,40,600.80630.820.69680.940.71780.9025,50,600.76850.860.69990.960.70230.8815,30,6020.95570.820.82990.840.88400.8220,40,600.84730.800.74110.920.77960.8625,50,600.76430.800.64940.940.72670.8415,30,6031.10490.780.92630.940.96100.8020,40,600.84150.760.75430.920.75980.8825,50,600.72070.840.65970.940.67050.9215,30,601.5011.02460.760.93320.920.93860.8620,40,600.80870.860.73250.900.75390.9025,50,600.77680.880.69861.000.72230.9215,30,6021.03560.760.98700.921.02600.8620,40,600.81690.840.70540.960.73210.8825,50,600.71700.760.65410.900.66480.8815,30,6030.99830.820.97860.900.98290.8820,40,600.86020.860.82370.940.82770.9025,50,600.74690.820.65840.920.68350.84Table 7
The values of AL and corresponding CP of 95% confidence interval for the ML and Bayesian estimates ofSt.
MLBayesianPrior 1Prior 2k,m,nTSchemeALCPALCPALCP15,30,600.5010.15450.960.14610.960.15050.9620,40,600.15020.900.14740.980.14860.9225,50,600.14680.880.14410.960.14420.9215,30,6020.14980.920.14540.980.14870.9620,40,600.14880.900.14340.980.14570.9425,50,600.15030.940.14830.980.14840.9815,30,6030.14950.940.14650.960.14760.9420,40,600.14790.900.14440.940.14520.9425,50,600.14790.940.14450.960.14480.9615,30,601.0010.17440.980.15841.000.15940.9820,40,600.15640.940.15220.980.15380.9625,50,600.15220.920.14940.920.14990.9215,30,6020.15980.900.15880.900.15900.9020,40,600.15430.940.15210.980.15260.9425,50,600.15390.880.15040.980.15210.9415,30,6030.15860.920.15630.940.15720.9220,40,600.15390.900.15090.940.15140.9225,50,600.14810.900.14530.920.14590.9215,30,601.5010.16460.880.16140.920.16270.9020,40,600.15890.940.15610.980.15640.9625,50,600.15200.900.14910.940.15170.9415,30,6020.16020.880.15200.900.15770.9020,40,600.15560.940.15280.960.15360.9425,50,600.15430.900.15040.960.15040.9415,30,6030.15810.940.15470.960.15480.9620,40,600.15290.940.15070.980.15080.9425,50,600.15040.960.14870.960.14900.96Table 8
The values of AL and corresponding CP of 95% confidence interval for the ML and Bayesian estimates ofHt.
MLBayesianPrior 1Prior 2k,m,nTSchemeALCPALCPALCP15,30,600.5010.34530.820.15720.940.18740.8420,40,600.32770.920.32240.960.32300.9225,50,600.30630.940.29990.960.30050.9415,30,6020.35530.860.35050.940.35160.8820,40,600.32940.860.32450.940.32510.9225,50,600.30950.960.30370.980.30500.9815,30,6030.35500.880.35320.940.35400.9420,40,600.32200.920.31460.980.31750.9825,50,600.30770.940.30190.960.30250.9415,30,601.0010.36230.960.16380.980.18660.9620,40,600.33870.960.18300.960.20180.9625,50,600.30870.920.18050.980.19030.9615,30,6020.38680.900.38440.920.38640.9220,40,600.33520.940.32910.980.32980.9825,50,600.31950.880.31440.940.31620.9215,30,6030.38560.900.37200.920.37440.9220,40,600.33890.980.33251.000.33331.0025,50,600.31780.920.31180.960.31240.9615,30,601.5010.37880.860.18780.940.21670.9020,40,600.34330.920.20230.940.21620.9425,50,600.30870.920.17760.940.19720.9415,30,6020.36600.880.35870.940.36380.9420,40,600.34300.960.33640.980.33820.9625,50,600.32690.940.32070.960.32090.9415,30,6030.37900.980.37101.000.37650.9820,40,600.33370.940.32801.000.32830.9425,50,600.31740.960.31160.980.31210.96To check the convergence of MCMC samples, we provide the key diagnostic test and trace plots with posterior density plots for different parameters and censoring schemes using the three above schemes atm=40 and T=1.5 with prior 1 and prior 2. Figures 1, 2, 3, and 4 show the trace plot iterations for posterior densities of α, λ, St, and ht, respectively. All censoring schemes are plotted in the same way, and it has been found that the trace plots of all censoring schemes converge very well.Figure 1
Trace plots of 11,000 iterations ofα.Figure 2
Trace plots of 11,000 iterations ofλ.Figure 3
Trace plots of 11,000 iterations ofSt=0.6.Figure 4
Trace plots of 11,000 iterations ofSt=0.6.From Tables1–4, as expected, we observe that the Bayesian estimates computed under informative prior have smaller MSEs than the Bayesian estimates computed under noninformative prior and ML estimates. Moreover, in most cases, the MSEs are decreases with increasing sample size m,k with the same timeT. Also, for the same sample sizes, the MSEs are decreases with increasing the time T. Furthermore, in most cases, under the scheme 1, MSEs are smaller than corresponding with scheme 2 and scheme 3.From Tables5–8 may be observed that the AL of Bayesian intervals with informative priors is shorter than the corresponding length of Bayesian intervals with noninformative priors; this is to be expected. As a result, if prior information is available, it should be used. Finally, there are no significance differences between the AL of ML intervals and the corresponding AL of Bayesian intervals with noninformative priors. Hence, we can say that performance of the maximum likelihood method is worse than the Bayesian method based on informative priors.


## 6. Real Data Analysis

In this section, we perform the following data analysis for illustrative purpose. The data set is from Lawless [24]. The data given here arose in tests on endurance of deep groove ball bearings. The data are the numbers of million revolutions before failure for each of the 23 ball bearings in the life test. It has been analyzed by several authors. It has been used earlier by Gupta and Kundu [14] that the two-parameter GE distribution can be used quite effectively to analyze this data set. Also, the Kolmogorov-Smirnov goodness-of-fit test with a total sample size of 23 was conducted for the GE distribution. The test statistics D=0.105825 and the corresponding p value of 0.93511 were obtained. As a result, the data can be considered fit to the GE distribution. We have created three Type I generalized PHCS from this uncensored data set, by fixed m=15, k=9, and R=2,07,3,05,3, and in different values of T, we have the censoring schemes as given in Table 9. The point and the 95% confidence interval estimates of the parameters α, λ, St=50, and ht=50 using the ML and informative and noninformative priors are presented in Table 10.Table 9
The different Type I generalized HPCS from the real data.
Scheme 1T=50D∗=9T∗=xk=54.12x¯=17.88,28.92,33.00,42.12,45.60,48.80,51.84,51.96,54.12R∗=2,0,0,0,0,0,0,0,12,Rτ∗=0Scheme 2T=80D∗=12T∗=Tx¯=17.88,28.92,33.00,42.12,45.60,48.80,51.84,51.96,54.12,55.56,67.80,68.88R∗=2,0,0,0,0,0,0,0,3,0,0,0,Rτ∗=6Scheme 3T=110D∗=15T∗=xm=105.84x¯=17.88,28.92,33.00,42.12,45.60,48.80,51.84,51.96,54.12,55.56,67.80,68.88,98.64,105.12,105.84R∗=2,0,0,0,0,0,0,0,3,0,0,0,0,0,3,Rτ∗=0Table 10
The point and 95% confidence interval estimates for the ML and Bayesian estimates ofα, λ, St=50, and Ht=50.
ParameterSchemeMLBayesianAsymp. CIBayesianPrior 1Prior 2Prior 1Prior 2αSch. 14.64363.97034.32392.3051,8.06012.6509,6.44812.4204,7.6571Sch. 24.41853.99364.08922.1934,8.47082.5224,6.77672.3030,8.0473Sch. 34.40803.88623.96742.1882,8.49092.5164,6.79272.2976,8.0664λSch. 10.02950.02820.02880.0143,0.04590.0152,0.04360.0146,0.0450Sch. 20.02900.02780.02840.0139,0.04340.0147,0.04120.0142,0.0425Sch. 30.02830.02710.02770.0121,0.04230.0128,0.04020.0123,0.0415St=50Sch. 10.70150.67060.68860.5772,1.07320.6350,1.01950.6119,1.0088Sch. 20.69240.68170.67730.5800,1.07220.6380,1.01860.6148,1.0079Sch. 30.70650.68700.68130.5758,1.07360.6333,1.02000.6103,1.0092Ht=50Sch. 10.01730.01780.01750.0095,0.02240.0105,0.02060.0106,0.0215Sch. 20.01740.01720.01760.0095,0.02240.0107,0.02080.0108,0.0213Sch. 30.01660.01670.01720.0096,0.02220.0103,0.02000.0100,0.0211


## 7. Conclusions and Discussion

In this paper, using the Type I progressive hybrid censored data from generalized exponential distribution, we construct the maximum likelihood and Bayesian estimators for the distribution’s parameters, and the maximum likelihood and Bayesian estimators of the reliability and hazard functions are computed. Using the delta technique, we determined the approximate confidence intervals of the reliability and hazard functions, as well as the approximate confidence intervals of the distribution’s unknown parameters. Finally, we used the Markov chain Monte Carlo approach to perform a Bayesian estimate procedure and determine credible intervals. The results showed that the Bayesian estimation is more reliable than the ML estimation.



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