# Inferences for Exponentiated Gamma Constant-Stress Partially Accelerated Life Test Model Based on Generalized Type-I Hybrid Censored Data

Authors: Abdalla Rabie; Abd-EL-Baset A. Ahmad; Thierno Souleymane Barry; Hassan M. Aljohani; Nada M. Alfaer; Abdulaziz S. Alghamdi
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

In this paper, the exponentiated gamma distribution (EGD) with generalized Type-I hybrid censored data under constant-stress partially accelerated life test (CSPALT) model is considered. The Bayesian and E-Bayesian estimation methods, as well as the maximum likelihood estimation method, are discussed for the parameter of the distribution and the acceleration factor. The E-Bayesian and Bayesian estimates are derived by using the squared error loss (SEL) and the LINEX loss functions. The MCMC method is applied for deriving the Bayesian and then E-Bayesian estimates. Moreover, a real data set is given for the illustrative purpose. After all, an evaluation is performed for the results of the proposed methods.


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## Body

## 1. Introduction

Many types of censoring schemes are used in the last decades. Type-I and Type-II censoring methods are still the most often used censoring schemes. In Type-I censoring, the test is terminated at a specified timeτ and the number of failures is random. In Type-II censoring, the test is ended after obtaining a prefixed number of failures while the time of the test is random. In these two types, the experimenter does not know when he will finish the test (as in Type-II) or he can get the required number before time τ (as in Type-I). To overcome these disadvantages of Type-I and Type-II, Epstein [1] introduced a mixture of Type-I and Type-II and referred to as hybrid censoring scheme (HCS) mainly, Type-I HCS and Type-II HCS. In Type-I HCS, the test is terminated at a random time T∗=minXr:n,T, where r∈1,2,…,n and T∈0,∞ are fixed from the begining of the test. Regarding Type-II HCS, the examination is terminated at a random point of time, and let us say it T∗=maxXr:n,T. Also, these schemes have drawbacks like having a few number of failures or not knowing the maximum time to finish the test. Therefore, Chandrasekar et al. [2] proposed an efficient and new censoring scheme which known as the generalized hybrid censoring schemes (generalized HCSs) to overcome drawbacks of HCSs. These schemes are considered as an extension of Type-I and Type-II HCSs. Therefore, we can notice that two types of censoring schemes are defined along these lines.Generalized Type-I HCS: letk,r∈1,2,…,n be integers and k<r<n, with time point T∈0,∞. In this scheme, the test is ended at minXr:n,T when the k-th failure happens before T. If the k-th failure is occurred next to T, the experiment is terminated at Xk:n. Generalized Type-II HCS: set r∈1,2,…,n with time points T1,T2∈0,∞ where T1<T2. If the r-th failure occurs sooner than time T1, the experiment is finished at T1. If the r-th failure is obtained between T1 and T2, the experiment is terminated at Xr:n. In the end, if the r-th failure occurs after time T2, the test is terminated at T2. These schemes are studied by many authors such as Huang and Yang [3]; Rabie, and Li [4]; and Rabie and Li [5].Due to the continuous enhancement in industrial design, it is difficult to get enough information of the products in the reliability tests under normal use conditions. Therefore, the accelerated life test (ALT) and partially accelerated life test (PALT) are used for this purpose. In these tests, a sample is subjected to more severe operating conditions than normal use conditions to obtain rapid failures. ALT items are allocated only in accelerated condition, while PALT is applied with normal and accelerated use conditions. This technique of tests results in shorter lifetimes than under normal use conditions. Extreme stress can be applied in several ways; the most common methods are step-stress and constant-stress as designated in Nelson [6]. In step-stress PALT, units first run at normal use condition; if units do not fail at a specified time, they allocated at accelerated use condition. Under constant stress, test items are divided into two groups: one of them runs at normal use conditions and the other is subjected to accelerated use conditions. We focus on the CSPALT model with generalized Type-I HCS in this paper.


### 1.1. The Model Description and Test Procedures

The exponentiated gamma distribution (EGD) was suggested by Gupta et al. [7] as an alternative to Weibull and gamma distributions. This study suggested that the EGD can present a better fit to the real data set than the GD. The cumulative distribution function (CDF), probability density function (PDF), and the reliability function (R(t)) of the EGD are written, respectively, in the forms as follows:(1)Fx;θ=1−e−xx+1θ,x>0,θ>0,(2)fx;θ=θxe−x1−e−xx+1θ−1,x>0,θ>0,(3)Rx;θ=1−1−e−xx+1θ,x>0,θ>0,where θ is the shape parameter. It is noted that when θ=1, the EGD turns into G2,1; for more details, one can refer to Shawky and Bakoban [8–10], Singh et al. [11]; Khan and Kumar [12]; Ghanizadeh et al. [13]; and Feroze and Aslam [14].Figure1 shows the plots of the shape of the PDF and CDF of the EGD distribution; it can be seen that the PDF has a unique mode as the parameter θ decreases. The distance between the shape of CDF increases as the parameter θ increases.Figure 1
Plots the shapes of the PDF (a) and CDF (b) of the EGD.
(a)(b)In CSPALT, a sample of sizen of test items is divided into two groups n1 and n2 chosen randomly among n items. n1 items run at normal use conditions, while n2 items are allocated at accelerated use conditions at the same time. The experiment is planned to continue at most until time T in both normal use and accelerated use conditions. We desire to obtain r1and r2 failures out of n1 and n2, respectively. And a bare minimum acceptable number of failures are k1 and k2 from n1 and n2, respectively. According to the CSPALT model under generalized Type-I HCS, one can note the following three cases of censored data. For normal use conditions, we observeCase I:X1:n1<X2:n1<⋯<Xk1:n1   if Xk1:n1>TCase II:X1:n1<⋯<Xk1:n1<⋯<Xr1:n1   if Xr1:n1<TCase III:X1:n1<⋯<Xk1:n1<⋯<Xd1:n1   if Xr1:n1>Twhere d1 denotes the number of failures occurring up to time T in the case of normal use conditions.Also, for accelerated use conditions, we observe the following three cases:Case 1:Y1:n2<Y2:n2<⋯<Yk1:n2   if Yk2:n2>TCase 2:Y1:n2<⋯<Yk2:n2<⋯<Yr2:n2   if Yr2:n2<TCase 3:Y1:n2<⋯<Yk2:n2<⋯<Yd2:n2   if Yr2:n2>Twhere Y=λ−1X, λ is the acceleration factor, and d2 stands for the number of failures occurring up to T in the case of accelerated use conditions.The remainder of this article is organised as follows: In Section2, the maximum likelihood based on CSPALT generalized Type-I HCS is discussed. In Section 3, the Bayesian estimation is studied under the SEL and LINEX loss functions by using the MCMC method. In Section 4, we present the Bayesian estimates based on the MCMC method. In Section 5, we present the E-Bayesian estimates based on the SEL and LINEX loss functions. In Section 6, we present the simulation study of the algorithm. We examine the flexibility of the distribution to fit the accelerated data in Section 7, so we provided a real data example, and the numerical results concluded are presented to asses the performance of the distribution.


## 1.1. The Model Description and Test Procedures

The exponentiated gamma distribution (EGD) was suggested by Gupta et al. [7] as an alternative to Weibull and gamma distributions. This study suggested that the EGD can present a better fit to the real data set than the GD. The cumulative distribution function (CDF), probability density function (PDF), and the reliability function (R(t)) of the EGD are written, respectively, in the forms as follows:(1)Fx;θ=1−e−xx+1θ,x>0,θ>0,(2)fx;θ=θxe−x1−e−xx+1θ−1,x>0,θ>0,(3)Rx;θ=1−1−e−xx+1θ,x>0,θ>0,where θ is the shape parameter. It is noted that when θ=1, the EGD turns into G2,1; for more details, one can refer to Shawky and Bakoban [8–10], Singh et al. [11]; Khan and Kumar [12]; Ghanizadeh et al. [13]; and Feroze and Aslam [14].Figure1 shows the plots of the shape of the PDF and CDF of the EGD distribution; it can be seen that the PDF has a unique mode as the parameter θ decreases. The distance between the shape of CDF increases as the parameter θ increases.Figure 1
Plots the shapes of the PDF (a) and CDF (b) of the EGD.
(a)(b)In CSPALT, a sample of sizen of test items is divided into two groups n1 and n2 chosen randomly among n items. n1 items run at normal use conditions, while n2 items are allocated at accelerated use conditions at the same time. The experiment is planned to continue at most until time T in both normal use and accelerated use conditions. We desire to obtain r1and r2 failures out of n1 and n2, respectively. And a bare minimum acceptable number of failures are k1 and k2 from n1 and n2, respectively. According to the CSPALT model under generalized Type-I HCS, one can note the following three cases of censored data. For normal use conditions, we observeCase I:X1:n1<X2:n1<⋯<Xk1:n1   if Xk1:n1>TCase II:X1:n1<⋯<Xk1:n1<⋯<Xr1:n1   if Xr1:n1<TCase III:X1:n1<⋯<Xk1:n1<⋯<Xd1:n1   if Xr1:n1>Twhere d1 denotes the number of failures occurring up to time T in the case of normal use conditions.Also, for accelerated use conditions, we observe the following three cases:Case 1:Y1:n2<Y2:n2<⋯<Yk1:n2   if Yk2:n2>TCase 2:Y1:n2<⋯<Yk2:n2<⋯<Yr2:n2   if Yr2:n2<TCase 3:Y1:n2<⋯<Yk2:n2<⋯<Yd2:n2   if Yr2:n2>Twhere Y=λ−1X, λ is the acceleration factor, and d2 stands for the number of failures occurring up to T in the case of accelerated use conditions.The remainder of this article is organised as follows: In Section2, the maximum likelihood based on CSPALT generalized Type-I HCS is discussed. In Section 3, the Bayesian estimation is studied under the SEL and LINEX loss functions by using the MCMC method. In Section 4, we present the Bayesian estimates based on the MCMC method. In Section 5, we present the E-Bayesian estimates based on the SEL and LINEX loss functions. In Section 6, we present the simulation study of the algorithm. We examine the flexibility of the distribution to fit the accelerated data in Section 7, so we provided a real data example, and the numerical results concluded are presented to asses the performance of the distribution.


## 2. The Likelihood Function Based on Constant-Stress Generalized Type-I Hybrid Censoring Scheme

We assume thatX1,X2,…,Xn1 are n1 observations of failure lifetimes under typical usage conditions that follows the generalized Type-I HCS, and Y1,Y2,…,Yn2 are n2 observations of breakdown lifetimes under accelerated usage conditions that follows the generalized Type-I HCS. The lifespan of test items is determined by EGD. So we can refer to the PDF under typical usage situation as in equation (2), and the following PDF is presented for a product in an accelerated consumption stage which can be written as the following:(4)fy;θ,λ=θλ2ye−λy1−e−λyλy+1θ−1,y,θ>0,λ>1,where Y=λ−1X. Therefore, if d1 denotes failures number obtained before T, so we can write the likelihood equation under generalized Type-I HCS for xj;θ,j=1,2,…,n1, without the multiplicative constant in usual usage is provided as(5)Lθ=θk11−φk1θn1−k1∏i=1k1xie−xiφiθ−1,d1=0,1,…,k1−1,θd11−φTθn1−d1∏i=1d1xie−xiφiθ−1,d1=k1,k1+1,…,r1−1,θr11−φr1θn1−r1∏i=1r1xie−xiφiθ−1,d1=r1,where φt=1−e−tt+1, and for yj;θ,λ;j=1,2,…,n2, if d2 indicates the number of failures that occur prior to the specified period T, we can refer to the likelihood equations without the multiplicative constant in accelerated usage which is provided as(6)Lθ,λ=θk2λ2k21−ψk2θn2−k2∏j=1k2yje−λyjψjθ−1,d2=0,1,…,k2−1,θd2λ2d21−ψTθn2−d2∏j=1d2yje−λyjψjθ−1,d2=k2,k2+1,…,r2−1,θr2λ2r21−ψr2θn2−r2∏j=1r2yje−λyjψjθ−1,d2=r2,where ψt=1−e−λtλt+1. By combining equations (5) and (6), the total likelihood function for xi;θ,yj;θ,λ:i=1,…,n1;j=1,…,n2 can be written as follows:(7)Lθ,λ=θk1+k2λ2k21−φk1θn1−k11−ψk2θn2−k2∏i=1k1xie−xiφiθ−1×∏j=1k2yje−λyjψjθ−1,d1=0,1,…,k1−1,d2=0,1,…,k2−1,θd1+d2λ2d21−φTθn1−d11−ψTθn2−d2∏i=1d1xie−xiφiθ−1×∏j=1d2yje−λyjψjθ−1,d1=k1,k1+1,…,r1−1,d2=k2,k2+1,…,r2−1,θr1+r2λ2r21−φr1θn1−r11−ψr2θn2−r2∏i=1r1xie−xiφiθ−1×∏j=1r2yje−λyjψjθ−1,d1=r1,d2=r2.


### 2.1. Maximum Likelihood Estimation Method

As it is stated, the log-likelihood is monotonically increasing so maximizing the likelihood function is equivalent to maximizing the log-likelihood. By calculating the log of the expression (7), as proceeds, we obtain the log-likelihood function:(8)ℓ=lnLθ,λ=k1+k2lnθ+2k2lnλ+n1−k1ln1−φk1θ+n2−k2ln1−ψk2θ+∑i=1k1lnxi−∑i=1k1xi+θ−1∑i=1k1lnφi+∑j=1k2lnyj−∑j=1k2λyj+θ−1∑j=1k2lnψj,d1=0,1,…,k1−1,d2=0,1,…,k2−1,d1+d2lnθ+2d2lnλ+n1−d1ln1−φTθ+n2−d2ln1−ψTθ+∑i=1d1lnxi−∑i=1d1xi+θ−1∑i=1d1lnφi+∑j=1d2lnyj−λ∑j=1d2yj+θ−1∑j=1d2lnψj,d1=k1,k1+1,…,r1−1,d2=k2,k2+1,…,r2−1,r1+r2lnθ+2r2lnλ+n1−r1ln1−φr1θ+n2−r2ln1−ψr2θ+∑i=1r1lnxi−∑i=1r1xi+θ−1∑i=1r1lnφi+∑j=1r2lnyj−λ∑j=1r2yj+θ−1∑j=1r2lnψj,d1=r1,d2=r2.Set the first derivatives of equation (8) regarding θ and λ to zero and solve the following equations numerically:(9)∂ℓ∂θ=0,∂ℓ∂λ=0,to get maximum likelihood estimates (MLEs) of θ and λ.


## 2.1. Maximum Likelihood Estimation Method

As it is stated, the log-likelihood is monotonically increasing so maximizing the likelihood function is equivalent to maximizing the log-likelihood. By calculating the log of the expression (7), as proceeds, we obtain the log-likelihood function:(8)ℓ=lnLθ,λ=k1+k2lnθ+2k2lnλ+n1−k1ln1−φk1θ+n2−k2ln1−ψk2θ+∑i=1k1lnxi−∑i=1k1xi+θ−1∑i=1k1lnφi+∑j=1k2lnyj−∑j=1k2λyj+θ−1∑j=1k2lnψj,d1=0,1,…,k1−1,d2=0,1,…,k2−1,d1+d2lnθ+2d2lnλ+n1−d1ln1−φTθ+n2−d2ln1−ψTθ+∑i=1d1lnxi−∑i=1d1xi+θ−1∑i=1d1lnφi+∑j=1d2lnyj−λ∑j=1d2yj+θ−1∑j=1d2lnψj,d1=k1,k1+1,…,r1−1,d2=k2,k2+1,…,r2−1,r1+r2lnθ+2r2lnλ+n1−r1ln1−φr1θ+n2−r2ln1−ψr2θ+∑i=1r1lnxi−∑i=1r1xi+θ−1∑i=1r1lnφi+∑j=1r2lnyj−λ∑j=1r2yj+θ−1∑j=1r2lnψj,d1=r1,d2=r2.Set the first derivatives of equation (8) regarding θ and λ to zero and solve the following equations numerically:(9)∂ℓ∂θ=0,∂ℓ∂λ=0,to get maximum likelihood estimates (MLEs) of θ and λ.


## 3. Bayesian Analysis

In this part of the paper, we made the most important estimation technique which is the Bayesian estimation for the two parametresθ and λ. We made estimation using diffirent estimation loss functions such as the SEL and LINEX loss functions. We assumed that the test follows CSPALT generalized Type-I hybrid censored sample from EGD. We uesd gamma prior for the two parameters θ and λ like gamma a1,a2 and gamma b1,b2, respectively. So, we can write the joint prior PDF equation of the two parameters of the distribution θ and λ, and it will have the following form:(10)πθ,λ=π1θπ2λ∝θa1−1λb1−1e−a2θ+b2λ,where(11)π1θ∝θa1−1exp−a2θ,a1,a2>0,θ>0,π2λ∝λb1−1exp−b2λ,b1,b2>0,λ>1.Then, the joint posterior PDF ofθ and λ is written from (7) and (10) as follows:(12)π∗θ,λ∣xi,yj=1Kθk1+k2+a1−1λ2k2+b1−11−φk1θn1−k11−ψk2θn2−k2∏i=1k1xie−xiφiθ−1×e−a2θ+b2λ∏j=1k2yje−λyjψjθ−1,d1=0,1,…,k1−1,d2=0,1,…,k2−1,1Kθd1+d2+a1−1λ2d2+b1−11−φTθn1−d11−ψTθn2−d2∏i=1d1xie−xiφiθ−1×e−a2θ+b2λ∏j=1d2yje−λyjψjθ−1,d1=k1,…,r1−1,d2=k2,…,r2−1,1Kθr1+r2+a1−1λ2r2+b1−11−φr1θn1−r11−ψr2θn2−r2∏i=1r1xie−xiφiθ−1×e−a2θ+b2λ∏j=1r2yje−λyjψjθ−1,d1=r1,d2=r2,where  K is a normalizing constant defined by(13)K=∫1∞∫0∞π∗θ,λ∣xi,yjdθdλ.Under the SEL function, the Bayesian estimate of any functionuθ,λ of θ and λ is given by(14)u^BS=Euθ,λ∣x,y=∫1∞∫0∞uθ,λπ∗θ,λ∣xi,yjdθdλ=1K∫1∞∫0∞uθ,λθk1+k2+a1−1λ2k2+b1−11−φk1θn1−k11−ψk2θn2−k2∏i=1k1xie−xiφiθ−1×e−a2θ+b2λ∏j=1k2yje−λyjψjθ−1dθdλ,d1=0,1,…,k1−1,d2=0,1,…,k2−1,1K∫1∞∫0∞uθ,λθd1+d2+a1−1λ2d2+b1−11−φTθn1−d11−ψTθn2−d2∏i=1d1xie−xiφiθ−1×e−a2θ+b2λ∏j=1d2yje−λyjψjθ−1dθdλ,d1=k1,…,r1−1,d2=k2,…,r2−1,1K∫1∞∫0∞uθ,λθr1+r2+a1−1λ2r2+b1−11−φr1θn1−r11−ψr2θn2−r2∏i=1r1xie−xiφiθ−1×e−a2θ+b2λ∏j=1r2yje−λyjψjθ−1dθdλ,d1=r1,d2=r2.Based on the LINEX loss function, the Bayesian estimate ofuθ,λ is given by(15)u^BL=−1hlnEe−huθ,λ∣x,y=−1hln∫1∞∫0∞e−huθ,λπ∗θ,λ∣xi,yjdθdλ=−1Khln∫1∞∫0∞θk1+k2+a1−1λ2k2+b1−11−φk1θn1−k11−ψk2θn2−k2∏i=1k1xie−xiφiθ−1×e−huθ,λ+a2θ+b2λ∏j=1k2yje−λyjψjθ−1dθdλ,d1=0,1,…,k1−1,d2=0,1,…,k2−1,−1Khln∫1∞∫0∞θd1+d2+a1−1λ2d2+b1−11−φTθn1−d11−ψTθn2−d2∏i=1d1xie−xiφiθ−1×e−huθ,λ+a2θ+b2λ∏j=1d2yje−λyjψjθ−1dθdλ,d1=k1,…,r1−1,d2=k2,…,r2−1,−1Khln∫1∞∫0∞θr1+r2+a1−1λ2r2+b1−11−φr1θn1−r11−ψr2θn2−r2∏i=1r1xie−xiφiθ−1×e−huθ,λ+a2θ+b2λ∏j=1r2yje−λyjψjθ−1dθdλ,d1=r1,d2=r2.It is clear from (14) and (15) that Bayesian estimates of θ and λ cannot be directly calculated, so the MCMC method is used for this purpose.


## 4. Bayesian Estimates Using MCMC Method

Here, we present the MCMC technique to compute and find the Bayesian estimates ofθ and λ. The conditional posterior PDF of the parameter θ and the acceleration factor λ is, respectively, written as follows:(16)π∗θ∣λ;xi,yj=θk1+k2+a1−11−φk1θn1−k11−ψk2θn2−k2∏i=1k1φiθ−1×e−a2θ∏j=1k2ψjθ−1,d1=0,1,…,k1−1,d2=0,1,…,k2−1,θd1+d2+a1−11−φTθn1−d11−ψTθn2−d2∏i=1d1φiθ−1×e−a2θ∏j=1d2ψjθ−1,d1=k1,…,r1−1,d2=k2,…,r2−1,θr1+r2+a1−11−φr1θn1−r11−ψr2θn2−r2∏i=1r1φiθ−1×e−a2θ∏j=1r2ψjθ−1,d1=r1,d2=r2,and(17)π∗λ∣θ;xi,yj=λ2k2+b1−11−ψk2θn2−k2e−b2λ∏j=1k2e−λyjψjθ−1,d1=0,1,…,k1−1,d2=0,1,…,k2−1,λ2d2+b1−11−ψTθn2−d2e−b2λ∏j=1d2e−λyjψjθ−1,d1=k1,…,r1−1,d2=k2,…,r2−1,λ2r2+b1−11−ψr2θn2−r2e−a2θ+b2λ∏j=1r2e−λyjψjθ−1,d1=r1,d2=r2.As it is seen from equations (16) and (17), the conditional posterior PDF of θ and λ does not look like any well-known models. Therefore, we use Metropolis-Hastings techinque to produce samples of θ and λ from the conditional posterior PDF using normal proposal distribution. Posterior samples of θ and λ are, respectively, generated from equations (16) and (17) using the Metropolis-Hastings algorithm.Step 1: First we initiate with starting values ofθ and λ and let it be the MLE values θ^MLE,λ^MLE.Step 2: takej=1.Step 3: from equation (16), generate θj and produce samples of θ∗ from a normal distribution as a proposal distribution.Step 4: Now we will compute the probability of accepting or rejecting the generated sample which callled the acceptance probability by using the following equation:(18)rθj−1∣θ∗=min1,π∗θ∗∣λj−1π∗θj−1∣λj−1.Step 5: we will produce samples from the uniform distribution ranged from zero to one as the following:U∼U0,1.Step 6: ifU≤rθj−1∣θ∗, we accept the generated value and assign θ∗=θj; else, reject the proposal and put θj−1=θj.Step 7: by the same way we will produce and generateλj by using equation (17) and generate λ∗using the normal distribution, and we consider it as the proposal distribution.Step 8: Make a repetition for the steps from step 4 to step 6 for the parameterλ too.Step 9: Assignj=j+1.Step 10: steps3−9 are repeated for N repetitions.Step 11: We can compute the Bayesian estimate values of the two parametersθ and λ using SEL function which are, respectively, as below:(19)θ^BS=1N−M∑j=M+1Nθj,(20)λ^BS=1N−M∑j=M+1Nλj,whereM is the number of iterations that are not considerd in the calculation, and sometime we call it nburn iterations.Step 12: We can compute the Bayesian estimates values of the two parametersθ and λ using LINEX loss function which are, respectively, as below:(21)θ^BL=−1hln1N−M∑j=M+1Ne−hθj,(22)λ^BL=−1hln1N−M∑j=M+1Ne−hλj.


## 5. E-Bayesian Estimation Method

The expectation of Bayesian estimation is referred to as “E-Bayesian estimation” and described as follows.Definition 1.
Letθ^a,b be continuous, then(23)δ^EB=Eδ^a,b=∬Qδ^a,bπa,bdadb,is called the expected Bayesian estimation of δ (briefly E-Bayesian estimation) where δ^a,b is the Bayesian estimate of δ with hyperparameters a and b, Q is the domain of a,b, and πa,b is the prior PDF of a and b over Q.From Han [15], the prior parameters (a1, a2) and (b1,b2) must be picked in order to ensure that π1θ and π2λare indeed a pair of declining functions of θ and λ, respectively. We can get the differentiation of π1θ regarding θ and π2λ with respect to λ as the following two equations:(24)dπ1θdθ∝θa1−2exp−a2θa1−1−a2θ,dπ2λdλ∝λb1−2exp−b2λb1−1−b2λ.When0<a1<1 and a2>0, dπ1θ/dθ<0, and when 0<b1<1 and b2>0, dπ2λ/dλ<0. Thus, π1θ and π2λ are decreasing functions for θ and λ, respectively. We make the assumption that the hyperparameters aj and bj, j=1,2, are independent and have the bivariate PDF given by(25)πa1,a2=π1a1π2a2,πb1,b2=π1b1π2b2.In order to get the E-Bayesian estimates of the two parametersθ and λ, we suggest the prior PDFs of (a1, a2) and (b1,b2) to clarify the impact of them on the E-Bayesian estimates of θ and λ. The prior PDFs of (a1, a2) and (b1,b2) are, respectively, given as follows:(26)π1a1,a2=2a1c1,0<a1<1,0<a2<c1,π2a1,a2=2a2c12,0<a1<1,0<a2<c1,π3a1,a2=3a22c13,0<a1<1,0<a2<c1,and(27)π1b1,b2=2b1c2,0<b1<1,0<b2<c2,π2b1,b2=2b2c22,0<b1<1,0<b2<c2,π3b1,b2=3b22c23,0<b1<1,0<b2<c2.


### 5.1. E-Bayesian Estimates Based on the Loss Functions

By substituting from (19) and (26) in (23), the E-Bayesian estimate of θ using the SEL function can be easily evaluated using the following equation:(28)θ^EB=EθB^a1,a2=∬QθB^a1,a2πia1,a2da1da2,i=1,2,3,and the E-Bayesian estimate of λ based on the SEL function is computed by substituting in (23) from (20) and (27) as follows:(29)λ^EB=Eλ^Bb1,b2=∬Qλ^Bb1,b2πib1,b2db1db2,i=1,2,3,where θ^Ba1,a2 and λ^Bb1,b2 are the estimates of the Bayesian method for θ and λ by applying the SEL function.Similarly, by substituting from (21) and (26) in (23), we get the E-Bayesian estimate of θ based on the LINEX loss function as follows:(30)θ^EB=Eθ^BLa1,a2=∬Qθ^BLa1,a2πia1,a2da1da2,i=1,2,3,and the E-Bayesian estimate of λ based on the LINEX loss function is obtained by substituting in (23) from (22) and (27) as follows:(31)λ^EB=Eλ^BLb1,b2=∬Qλ^BLb1,b2πib1,b2db1db2,i=1,2,3,where θ^BLa1,a2 and λ^BLb1,b2 are the estimates of θ and λ regarding the Bayesian method under the LINEX loss function. For details, one can see, Han [16]; Jaheen and Okasha [17]; Okasha [18]; Rabie and Li [19]; Rabie and Li [20]; and Rabie [21]. Before progressing, we describe procedures of the simulation study used in this paper.


## 5.1. E-Bayesian Estimates Based on the Loss Functions

By substituting from (19) and (26) in (23), the E-Bayesian estimate of θ using the SEL function can be easily evaluated using the following equation:(28)θ^EB=EθB^a1,a2=∬QθB^a1,a2πia1,a2da1da2,i=1,2,3,and the E-Bayesian estimate of λ based on the SEL function is computed by substituting in (23) from (20) and (27) as follows:(29)λ^EB=Eλ^Bb1,b2=∬Qλ^Bb1,b2πib1,b2db1db2,i=1,2,3,where θ^Ba1,a2 and λ^Bb1,b2 are the estimates of the Bayesian method for θ and λ by applying the SEL function.Similarly, by substituting from (21) and (26) in (23), we get the E-Bayesian estimate of θ based on the LINEX loss function as follows:(30)θ^EB=Eθ^BLa1,a2=∬Qθ^BLa1,a2πia1,a2da1da2,i=1,2,3,and the E-Bayesian estimate of λ based on the LINEX loss function is obtained by substituting in (23) from (22) and (27) as follows:(31)λ^EB=Eλ^BLb1,b2=∬Qλ^BLb1,b2πib1,b2db1db2,i=1,2,3,where θ^BLa1,a2 and λ^BLb1,b2 are the estimates of θ and λ regarding the Bayesian method under the LINEX loss function. For details, one can see, Han [16]; Jaheen and Okasha [17]; Okasha [18]; Rabie and Li [19]; Rabie and Li [20]; and Rabie [21]. Before progressing, we describe procedures of the simulation study used in this paper.


## 6. Simulation Study

Here, we provide the simulation results according to the following steps:Specify the values ofn,r1,r2,k1,k2,h,a1,a2,b1,b2,c1,c2,andT.First we use the uniform distribution to genrate a random samplen from U0,1.We will choosen1 items that are selected randomly, from n items, and subject them to the normal usage situations.Computen2=n−n1. These units, allocations, are subjected to stress conditions.Indicate the initial values used in generating data for the two paramtersθ and λ.Now we will produce a generalized Type-I hybrid censored sample from the EGD CSPALT model using the inverse function method by solvingU=1−e−xx+1θ regarding x for a typically use situations and by solving U=1−e−λyλy+1θ with respect to y, for stress use situations.We build a Markov chain containing 11,000 data by using the Metropolis–Hastings algorithm ofθ and λ, and we will not consider the first 1000 values as they are very affected with the initial values.θ^BS and λ^BS are the Bayesian estimation of the two parameters using the SEL function which can be easily computed by using equation (19) and equation (20), respectively.θ^BL and λ^BL are the E-Bayesian estimation of the two parameters using the LINEX loss function which can be easily computed by using equation (21) and equation (22), respectively.θ^EBS and λ^EBSare the E-Bayesian estimation of the two parameters using the SEL function which can be easily computed by using equation (28) and equation (29), respectively.θ^EBL and λ^EBLare the E-Bayesian estimation of the two parameters using the LINEX loss function which can be easily computed by using equation (30) and (31), equation respectively.We can compute the mean squared error (MSE) or the estmiated values of the two parametersθ and λ, which are, respectively, as follows:(32)MSEθ^=11000∑j=11000θj^−θ2,MSEλ^=11000∑j=11000λj^−λ2,whereθ^ is considerd as the estimated values for θ and λ^ is considerd as the estimated values for λ.The numerical outcomes are obtained using MATHEMATICA 8 functions such as FindRoot, NMaximize, NIntegrate, and RandomReal and displayed in Tables1 and 2. Table 1 contains the numerical values of the Bayesian, E-Bayesian estimates, also it contains the MLEs, beside these values there is MSE of the parameter θ, using the LINEX and SEL functions. Table 2 contains the numerical values of the Bayesian, E-Bayesian estimates, also it contains the MLEs, beside these values there is MSE of the parameter λ using LINEX and SEL functions. By observing the results in Tables 1 and 2, we may infer that the E-Bayesian estimate outperforms the Bayesian estimator for the SEL and LINEX loss functions for the two parameters θ and λ as we can easily see it having small values for the MSE. Additionally, when the sample size rises, the MSE of Bayesian and E-Bayesian estimators decreases for a sample of size n and the censoring time T get increased.Table 1
Average estimates, MSE of MLEs, Bayesian, and E-Bayesian estimates forθ under LINEX and SEL at θ=0.7,λ=1.2,a1=b1=0.7,a2=1.4,b2=1.2,h=1.5,andc1=c2=2.
n1,n2TCriteriaθ^MLESquared error lossLINEX lossr1,r2k1,k2θ^BSθ^EBS1θ^EBS2θ^EBS3θ^BLθ^EBL1θ^EBL2θ^EBL320,252Mean0.753 9971.143 540.672 4020.576 3450.518 711.129 740.667 4090.572 6410.515 693(15, 20)(10, 15)MSE0.042 9220.210 20.005 4190.018 7120.035 6380.195 270.005 0920.019 2410.036 4462.5Mean0.772 421.060 930.623 8240.534 7060.581 1991.058 550.622 9860.534 0880.574 801MSE0.033 4190.130 290.005 8110.027 3280.017 5940.128 580.005 9390.027 5330.018 68530,402Mean0.926 181.045 550.614 7840.526 9580.474 2621.044 070.614 2610.526 5720.473 949(20, 30)(15, 20)MSE0.086 350.119 410.007 2640.029 9450.050 9590.118 390.007 3530.030 0780.051 12.5Mean0.818 9581.026 420.603 5380.517 3180.465 5861.025 980.603 3840.517 2050.465 494MSE0.019 730.106 560.009 3060.033 3730.054 950.106 270.009 3360.033 4150.054 99340,502Mean0.885 6941.035 890.609 1010.522 0870.469 8781.034 770.608 7030.521 7920.469 639(30, 40)(20, 30)MSE0.060 4160.113 040.008 3380.031 7080.053 0010.112 260.008 4050.031 810.053 1092.5Mean0.857 9841.010 30.594 0570.509 1920.458 2731.010 230.594 0310.509 1720.458 257MSE0.060 3040.096 290.011 2250.036 4080.058 4330.096 240.011 230.036 4160.058 44Table 2
Average estimates, MSE of MLEs, Bayesian, and E-Bayesian estimates forλ under LINEX and SEL at θ=0.7,λ=1.2,a1=b1=0.7,a2=1.4,b2=1.2,h=1.5,andc1=c2=2.
n1,n2TCriteriaλ^MLESquared error lossLINEX lossr1,r2k1,k2λ^BSλ^EBS1λ^EBS2λ^EBS3λ^BLλ^EBL1λ^EBL2λ^EBL320,252Mean1.011 881.559 981.070 151.070 151.123 651.525 091.053 511.053 511.105 34(15, 20)(10, 15)MSE0.123 5640.181 310.041 2030.041 2030.032 6650.151 840.043 9670.043 9670.033 6792.5Mean1.002 441.555 651.067 171.067 171.156 641.520 021.050 21.050 21.137 03MSE0.051 5840.133 960.021 1630.021 1630.015 5520.108 920.025 6390.025 6390.016 70530,402Mean1.126 311.691 171.160 141.160 141.218 151.663 991.147 251.147 251.203 95(20, 30)(15, 20)MSE0.044 1120.244 340.003 0430.003 0430.001 9330.218 260.004 1990.004 1990.001 5752.5Mean0.949 7651.511 481.036 871.036 871.088 721.490 41.026 891.026 891.077 72MSE0.062 930.104 280.030 0280.030 0280.016 1520.091 080.033 2160.033 2160.018 52540,502Mean1.087 321.552 711.065 161.065 161.118 421.534 221.056 41.056 41.108 77(30, 40)(20, 30)MSE0.045 6260.133 460.022 4420.022 4420.011 3530.120 310.024 7340.024 7340.012 8512.5Mean1.034 611.469 11.007 81.007 81.058 191.452 110.999 7740.999 7741.049 35MSE0.038 8130.075 080.038 1950.038 1950.021 4920.066 120.041 3120.041 3120.024 042


## 7. Example of Real-Life Data

In this section, an example of real-life data is provided to investigate the performance of the proposed methods in the application. These data were used by Singh et al. [22], representing the average monthly rainfall obtained from the Information System for Management of Water Resources from the State of So Paulo, including a period of 56 years from 1947 to 2003. Also, they checked the fitting of the given data set through different method of estimation and stated that EGD gives a good fit for these data. These data contain 56 observations listed as follows: 0.2,3.5,2.8,3.7,8.7,6.9,7.4,0.8,4.8,2.5,2.9,3.1,4.0,5.0,3.8,3.5,5.4,3.3,2.9,1.7,7.3,2.9,4.6,1.1,1.4,3.9,6.2,4.1,10.8,3.8,7.3,1.8,6.7,3.5,3.2,5.2,2.8,5.2,5.4,2.2,9.9,2.1,4.7,5.5,2.6,4.1,5.4,5.5,2.1,1.9,8.8,1.3,24.1,5.4,6.2,2.9.We suppose that values of data set represent lifetime of failure observations which follow the EGD. Now we will apply the CSPALT when the sample is genralized Type-I hybrid censoring scheme. Such thatn1=26 and n2=30, where the first sample and the second sample were selected randomly from the complete sample of size n=56 units. We desire to obtain r1=21 of failures out of n1=26 units, and k1=15 is a bare minimum number of failures that can be accepted out of n1=26 units. While we desire to obtain r2=25 failures out of n2=30 units, k2=20 represents a minimum number of failures is acceptable from n2=30 units. All estimates of θ and λ are derived based on the same previous procedures and shown in Tables 3 and 4. Table 3 gives estimates and MSE for the parameter θ of ML, Bayesian, and E-Bayesian estimation methods based on SEL and LINEX loss functions. Table 4 gives the previous criteria for the acceleration factor λ. By observing results listed in Tables 3 and 4 regarding the real data set, one can note that the E-Bayesian method is the best compared with both ML and Bayesian estimation methods because of having less MSE. Moreover, the proposed methods are easily applied to the real data and gave good results.Table 3
Real data set: average estimates, MSE of MLEs, Bayesian, and E-Bayesian estimates forθ under LINEX and SEL functions when θ=2,λ=1.24,a1=0.8,b1=0.7,a2=1.6,b2=1.4,h=1.5,andc1=c2=2.
n1,n2TCriteriaθ^MLESquared error lossLINEX lossr1,r2k1,k2θ^BSθ^EBS1θ^EBS2θ^EBS3θ^BLθ^EBL1θ^EBL2θ^EBL326,304.8Mean2.567 42.672 821.833 551.833 551.925 232.624 821.811 651.811 651.901(21, 25)(15, 20)MSE0.321 9450.452 720.027 7220.027 7220.005 6090.390 450.035 4950.035 4950.009 82426,305.8Mean2.578 212.679 141.837 891.837 891.929 782.632 41.816 581.816 581.906 21(21, 25)(15, 20)MSE0.334 3290.461 270.026 2990.026 2990.004 9510.399 980.033 6650.033 6650.008 821Table 4
Real data set: average estimates, MSE of MLEs, Bayesian, and E-Bayesian estimates forλ under LINEX and SEL functions when θ=2,λ=1.24,a1=0.8,b1=0.7,a2=1.6,b2=1.4,h=1.5,andc1=c2=2.
n1,n2TCriteriaλ^MLESquared error lossLINEX lossr1,r2k1,k2λ^BSλ^EBS1λ^EBS2λ^EBS3λ^BLλ^EBL1λ^EBL2λ^EBL326,304.8Mean1.027 721.062 141.087 631.087 631.305 151.059 941.085 331.085 331.301 87(21, 25)(15, 20)MSE0.045 0630.031 640.023 2260.023 2260.023 2260.004 2580.023 9320.023 9320.003 8426,305.8Mean1.029 651.062 481.087 981.087 981.305 581.060 271.085 661.085 661.302 27(21, 25)(15, 20)MSE0.044 2460.031 520.023 120.023 120.004 3160.032 310.023 8290.023 8290.003 891


## 8. Conclusion

In this paper, we studied the exponentiated gamma distribution (EGD) with generalized Type-I hybrid censored data under the constant-stress partially accelerated life test (CSPALT) model. We discussed the Bayesian and E-Bayesian estimation methods, as well as the maximum likelihood method, for the distribution parameter and the acceleration factor. The E-Bayesian and Bayesian estimates are obtained by the SEL and the LINEX loss functions. The MCMC method is used for deriving the Bayesian estimates, and then we computed the E-Bayesian estimates. We provided a real data set to clarify the behavior of the methods in the application.From the results shown in Tables1–4, we may conclude that the E-Bayesian estimation approach is superior to both ML and Bayesian estimation methods due to its lower MSE. Also, the E-Bayesian estimation method is easy to be applied and convenient to the application. Additionally, by including additional failure items in the CSPALT model with censoring strategies, adequate information about test units is obtained. Additionally, it is shown that the presented methodologies are simply applicable to the CSPALT model and provide acceptable results.



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