# Estimations in a Constant-Stress Partially Accelerated Life Test for Generalized Rayleigh Distribution under Type-II Hybrid Censoring Scheme

Authors: Abdalla Rabie; Eslam Hussam; Abdisalam Hassan Muse; Ramy Abdelhamid Aldallal; Amirah Saeed Alharthi; Hassan M. Aljohani
Journal: Journal of Mathematics
 (2022)

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## Abstract

The constant-stress partially accelerated life test (CSPALT) model with Type-II hybrid censoring scheme (Type-II HCS) is the subject of our research. Units have a lifetime that follows the generalized Rayleigh distribution. Bayesian and E-Bayesian estimates are derived by applying two of the loss functions, mainly the squared error loss (SEL) and LINEX loss functions. Bayesian and E-Bayesian estimates are obtained using Markov chain Monte Carlo (MCMC) methods. To prove the applicability and the importance of the subject, a test for real data will be provided. To evaluate the distribution’s effectiveness, we utilized a variety of datasets and proposed several kinds of censoring. Finally, all results are compared in order to determine the effectiveness of the proposed methods. All major findings are concluded in the conclusion section.


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## Body

## 1. Introduction

The majority of modern items is built to work efficiently over an extended length of time. So, the reliability tests in the normal use condition need more time and cost. In such instances, accelerated life tests (ALTs) are performed in order to collect more information for the reliability of the products in the shortest period. An increase in usual performance, temperature, pressure, stress, or any combination of these components constitutes an accelerated test condition. Regarding ALT, it is presumed that the accelerating factor is known, and the relationship between life span and stress is identified using a mathematical model. These models may not exist or be extremely difficult to construct in some circumstances. As a result, in this case, it is advisable to employ partially accelerated life tests (PALTs), which are a form of ALT. PALT is also utilized when only a specific acceleration condition is required, with the data being extrapolated to the test case. Nelson [1] demonstrated that frequent forms of stress, such as step stress and constant stress, exist. CSPALT, which divides test subjects into two groups before analyzing the data, will be discussed. Group 1 is for the regular case, while Group 2 is for the accelerated case. We will continue performing the experiment until all of the units fail or the time limit for the test has expired. To learn more, see [2–5].HCS is a censoring technique that combines Type-I and Type-II censoring. Type-I HCS, based on the time at which the test is terminated at the instant ofT∗=minYk:n,τ, such that Yk:n refers to the kth item failure time, and τ, can be defined as the full maximum time for the experiment under consideration. We ended the test when we reached kfailed items out of n items where k and τ are prefixed from the beginning, so we end the test at the time whether k or τ has been reached, whichever is closer, while in Type-II HCS, after a random period of time, the life-testing experiment is completed. Let us say at T∗=maxYk:n,τ, i.e., when the latter of two stopping rules is met, the experiment is halted. As a result, we may be certain that at least k failures have occurred. We assume in this study that the data are Type-II hybrid censored using the generalized Rayleigh CSPALT model, which indicates that each of the following cases may be observed.Under normal operating conditions:Case I:T1<T2<⋯<Tr1   if Tr1>τ.Case II:T1<T2<⋯<Tm1   if Tm1≤τ<Tm1+1.In the event of an accelerated use condition:Case 1:X1<X2<⋯<Xr2   if Xr2>τ.Case 2:X1<X2<⋯<Xm2   if Xm2≤τ<Xm2+1.Here,m1 and m2, are the number of failures recorded before time τ in regular and accelerated use settings, respectively.This article is divided into several sections. Section2 includes an explanation of the model and the test procedures. Section 3 discusses the maximum likelihood technique. Section 4 discusses Bayesian estimation using LINEX and SEL loss functions. Section 5 discusses the E-Bayesian technique for loss functions SEL and LINEX. Section 6 discusses the MCMC approach. Section 7 illustrates real datasets and concludes the findings of the suggested approaches.


### 1.1. Notations

n: all experiment items n=n1+n2.τ: the experiment’s maximum duration.T∗: termination time of the experiment’s real-life testing.X: product’s life expectancy under accelerated usage conditions.T: the duration of an item’s life span under typical usage conditions.xi: the time at which the item i fails which is put under accelerated use condition.ti: the time at which the item i fails which is put under normal use condition.n1,n2: Group 1 is assigned to typical usage, whereas Group 2 is assigned to stressed usage working conditions.θ: accelerating factor θ>1.r1: these are considered as the failed items from the n1 units before censoring time τ.r2: these are considered as the failed items from the n2 units after censoring time τ.Tj,j=1,2,…,n1: lifetimes of failure under typical use.Xi,i=1,2,…,n2: failure’s life expectancy under accelerated usage conditions.


## 1.1. Notations

n: all experiment items n=n1+n2.τ: the experiment’s maximum duration.T∗: termination time of the experiment’s real-life testing.X: product’s life expectancy under accelerated usage conditions.T: the duration of an item’s life span under typical usage conditions.xi: the time at which the item i fails which is put under accelerated use condition.ti: the time at which the item i fails which is put under normal use condition.n1,n2: Group 1 is assigned to typical usage, whereas Group 2 is assigned to stressed usage working conditions.θ: accelerating factor θ>1.r1: these are considered as the failed items from the n1 units before censoring time τ.r2: these are considered as the failed items from the n2 units after censoring time τ.Tj,j=1,2,…,n1: lifetimes of failure under typical use.Xi,i=1,2,…,n2: failure’s life expectancy under accelerated usage conditions.


## 2. Test Procedure and Model Description

### 2.1. The Model of Test Units Lifetime

The lifetimes of the test units are following the generalized Rayleigh model. The generalized Rayleigh (Burr-X) distribution is one of the Burr distributions, proposed by Burr [6]. It is critical in a variety of disciplines, including economics and computational modeling, as well as in health, agriculture, and biology. Here is the probability density function (PDF) of the generalized Rayleigh distribution, which is defined as follows:(1)ft;λ=2λte−t21−e−t2λ−1,t>0,λ>0,and the cumulative distribution function (CDF) is given by(2)Ft;λ=1−e−t2λ,t>0,λ>0,where λ is the shape parameter.The generalized Rayleigh reliability functionR(t) has the form (3)Rt=1−1−e−t2λ,and the hazard rate function ht is given by(4)ht=2λte−t21−e−t2λ−11−1−e−t2λ.Recently, many studies have been conducted on generalized Rayleigh distribution, for example, [7–11].


### 2.2. Test Procedure

All units of sizen in CSPALT will be separated into two categories, n1 and n2, as follows:(1)
n1 items run normally.(2)
n2 are randomly picked from n items and located under accelerated use conditions.(3)
At a random time,T∗, the test will be ended,where(5)T∗=maxTr1,τ,for regular usage condition,maxXr2,τ,due to the fact that utilisation is accelerated,and Tr1 is the failure time of r1th unit in the event of a typical operating state and Xr2 is the failure period of r2th units when the items were exposed to stress usage conditions.


## 2.1. The Model of Test Units Lifetime

The lifetimes of the test units are following the generalized Rayleigh model. The generalized Rayleigh (Burr-X) distribution is one of the Burr distributions, proposed by Burr [6]. It is critical in a variety of disciplines, including economics and computational modeling, as well as in health, agriculture, and biology. Here is the probability density function (PDF) of the generalized Rayleigh distribution, which is defined as follows:(1)ft;λ=2λte−t21−e−t2λ−1,t>0,λ>0,and the cumulative distribution function (CDF) is given by(2)Ft;λ=1−e−t2λ,t>0,λ>0,where λ is the shape parameter.The generalized Rayleigh reliability functionR(t) has the form (3)Rt=1−1−e−t2λ,and the hazard rate function ht is given by(4)ht=2λte−t21−e−t2λ−11−1−e−t2λ.Recently, many studies have been conducted on generalized Rayleigh distribution, for example, [7–11].


## 2.2. Test Procedure

All units of sizen in CSPALT will be separated into two categories, n1 and n2, as follows:(1)
n1 items run normally.(2)
n2 are randomly picked from n items and located under accelerated use conditions.(3)
At a random time,T∗, the test will be ended,where(5)T∗=maxTr1,τ,for regular usage condition,maxXr2,τ,due to the fact that utilisation is accelerated,and Tr1 is the failure time of r1th unit in the event of a typical operating state and Xr2 is the failure period of r2th units when the items were exposed to stress usage conditions.


## 3. Maximum Likelihood Method

We assume thatT1,T2,…,Tn1 are n1 failure lifespans observed in the state of typical use under Type-II HCS, and X1,X2,…,Xn2 are n2 observed lifetimes of failures supposing that the units are exposed to the accelerated use working situation under Type-II HCS. The lifespans of test items follow a generalized Rayleigh distribution. The PDF of a product in its normal-use state is specified in (1). The PDF for an item in the accelerated usage state is as follows:(6)fx;λ,θ=2λθ2xe−θx21−e−θx2λ−1,x>0,λ>0,θ>1,where X=θ−1T. As a result, for tj;λ,j=1,2,…,n1, at ordinary working condition by removing the constant, we may express the likelihood function in the following manner, under Type-II HCS:(7)Ltj;λ=∏j=1rλtje−tj21−e−tj2λ−1×1−1−e−s2λn1−r,where(8)r,s=r1,tr1,for Case I,m1,τ,for Case II.Similarly, forxi;λ,θ;i=1,2,…,n2, for accelerated usage, the likelihood function has the following format:(9)Lxi;λ,θ=∏i=1pλθ2xie−θxi21−e−θxi2λ−1×1−1−e−θd2λn2−p,where(10)p,d=r2,xr2,for Case1,m2,τ,for Case2.The function representing the total likelihood function fortj;λ,xi;λ,θ:j=1,…,n1;i=1,…,n2 is obtained by combining equations (7) and (9) as(11)Lλ,θ=λr+pθ2p1−1−e−s2λn1−r×1−1−e−θd2λn2−p×∏i=1pxie−θxi21−e−θxi2λ−1×∏j=1rtje−tj21−e−tj2λ−1.Applying the natural logarithm to equation (11) yields the following formula referred to as “the log-likelihood function”:(12)ℓ=lnLλ,θ=λ−1∑i=1pln1−e−θxi2+∑j=1rlntj+∑j=1r−tj2+r+plnλ+n1−rln1−1−e−s2λ+λ−1∑j=1rln1−e−tj2+∑i=1p−θxi2+lnxi+n2−pln1−1−e−θd2λ+2plnθ.The MLEs for the unknown parameterλ and the acceleration factor θ are obtained by equating the initial partial derivatives of equation (12) to zero with regard to both parameters and solving the following equations numerically.(13)∂ℓ∂λ=0,∂ℓ∂θ=0,and then the MLEs λ^ and θ^ of λ and θ are obtained.


## 4. Bayesian Estimation Approach

Based on the SEL and LINEX loss functions, Bayesian estimates ofλ and θ are derived in this part of study. CSPALT is used with a generalized Rayleigh Type-II hybrid censored sample. Gamma a1,b1 and Gamma a2,b2 are suggested as the prior PDFs of λ and θ, respectively. As a result, the combined prior PDF of λ and θ is written as(14)πλ,θ=π1λπ2θ,∝λa1−1θa2−1e−b1λ+b2θ,a1,a2,b1,b2>0,where(15)π1λ∝λa1−1e−b1λ,(16)π2θ∝θa2−1e−b2θ.We can write the posterior PDF of the two parametersλ and θ with the aid of equations (11) and (14) as follows:(17)π∗λ,θ=K−1λr+p+a1−1θ2p+a2−11−1−e−s2λn1−r×e−b1λ+b2θ∏i=1pxie−θxi21−e−θxi2λ−1×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p,where   a normalizing constant K is given by(18)K=∫1∞∫0∞π∗λ,θdλdθ.


### 4.1. Bayesian Estimates under SEL Function

Using the SEL function as an example, the posterior mean is used to estimateλ, as follows:(19)λ^BS=Eλ|t,x=∫1∞∫0∞λπ∗λ,θdλdθ=K−1∫1∞∫0∞λr+p+a1θ2p+a2−1e−b1λ+b2θ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ.Likewise, we may construct estimates for the other parameterθ using the same techniques as follows:(20)θ^BS=Eθ|t,x=∫1∞∫0∞θπ∗λ,θdλdθ=K−1∫1∞∫0∞λr+p+a1−1θ2p+a2e−b1λ+b2θ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ.


### 4.2. Bayesian Estimates under LINEX Loss Function

The Bayesian estimates ofλ and θ using the LINEX loss function are as follows:(21)λ^BL=−1hlnEe−hλ|t,x=−1hln∫1∞∫0∞e−hλπ∗λ,θdλdθ=−1Khln∫1∞∫0∞λr+p+a1−1θ2p+a2−1e−b1+hλ−b2θ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ,h≠0,(22)θ^BL=−1hlnEe−hθ|t,x=−1hln∫1∞∫0∞e−hθπ∗λ,θdλdθ=−1Khln∫1∞∫0∞λr+p+a1−1θ2p+a2−1e−b1λ−b2+hθ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ,h≠0.


## 4.1. Bayesian Estimates under SEL Function

Using the SEL function as an example, the posterior mean is used to estimateλ, as follows:(19)λ^BS=Eλ|t,x=∫1∞∫0∞λπ∗λ,θdλdθ=K−1∫1∞∫0∞λr+p+a1θ2p+a2−1e−b1λ+b2θ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ.Likewise, we may construct estimates for the other parameterθ using the same techniques as follows:(20)θ^BS=Eθ|t,x=∫1∞∫0∞θπ∗λ,θdλdθ=K−1∫1∞∫0∞λr+p+a1−1θ2p+a2e−b1λ+b2θ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ.


## 4.2. Bayesian Estimates under LINEX Loss Function

The Bayesian estimates ofλ and θ using the LINEX loss function are as follows:(21)λ^BL=−1hlnEe−hλ|t,x=−1hln∫1∞∫0∞e−hλπ∗λ,θdλdθ=−1Khln∫1∞∫0∞λr+p+a1−1θ2p+a2−1e−b1+hλ−b2θ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ,h≠0,(22)θ^BL=−1hlnEe−hθ|t,x=−1hln∫1∞∫0∞e−hθπ∗λ,θdλdθ=−1Khln∫1∞∫0∞λr+p+a1−1θ2p+a2−1e−b1λ−b2+hθ×∏j=1rtje−tj21−e−tj2λ−11−1−e−θd2λn2−p×1−1−e−s2λn1−r∏i=1pxie−θxi21−e−θxi2λ−1dλdθ,h≠0.


## 5. E-Bayesian Estimation Method

Refer to [12], the hyperparameters (a1, b1) and a2,b2 are selected such that π1λ and π2θ are decreasing functions.. We must obtain the derivatives of π1λ and π2θ in terms of λ and θ which are given by(23)dπ1λdλ∝λa1−2exp−b1λa1−1−b1λ,dπ2θdθ∝θa2−2exp−b2θa2−1−b2θ.When0<a1<1 and b1>0, dπ1λ/dλ<0, and when 0<a2<1 and b2>0, dπ2θ/dθ<0. Thus, π1λ and π2θ are diminishing functions for λ and θ, respectively. For the independent hyperparameters, we hypothesize aj and bj, j=1,2, bivariate PDFs provided by(24)πaj,bj=π1ajπ2bj,j=1,2.According to E-Bayesian estimations, the estimates of the distribution parameters can be computed with the aid of the following equations:(25)λ^EB=EλB^a1,b1=∬λB^a1,b1πia1,b1da1db1,i=1,2,3,(26)θ^EB=Eθ^Ba2,b2,=∬θ^Ba2,b2πia2,b2da2db2,i=1,2,3,where λ^Ba1,b1 and θ^Ba2,b2 stand for Bayesian estimated values of λ and θ under the effect of both SEL function and the LINEX loss function. For further information, we refer to [13–24].


### 5.1. E-Bayesian Estimate ofλ

To derive E-Bayesian estimates ofλ and θ, we consider the following prior PDFs of (a1, b1) and a2,b2 to clarify the impact of prior PDFs on the estimates of E-Bayesian method for λ and θ. The prior PDFs are given by(27)π1aj,bj=2ajcj,0<aj<1,0<bj<cj,π2aj,bj=2bjcj2,0<aj<1,0<bj<cj,π3aj,bj=3bj2cj3,0<aj<1,0<bj<cj,j=1,2.The E-Bayesian estimated value ofλ is obtained using the SEL function from (19), (25), and (27) as follows:(28)λ^EBSi=∬λ^BSa1,b1πia1,b1da1db1,i=1,2,3.Additionally, the E-Bayesian estimate ofλ may be produced through using LINEX loss function from (21), (25), and (27) as follows:(29)λ^EBLi=∬λ^BLa1,b1πia1,b1da1db1,i=1,2,3.


### 5.2. E-Bayesian Estimate ofθ

Based on the SEL and LINEX loss functions, the E-Bayesian estimate is obtained by using (27).We can use (20), (26), and (27) to obtain the estimates of θ as follows:(30)θ^EBSi=∬θ^BSπia1,b1da2db2,i=1,2,3,and for the LINEX loss function, we can use (22), (26), and (27) to obtain the estimate of the same parameter:(31)θ^EBLi=∬θ^BLπia1,b1da2db2,i=1,2,3.We employ the MCMC methodology to generate Bayesian and E-Bayesian estimates ofλ and θ because all estimators of Bayesian and E-Bayesian methods cannot be expressed explicitly as shown in the preceding equations.


## 5.1. E-Bayesian Estimate ofλ

To derive E-Bayesian estimates ofλ and θ, we consider the following prior PDFs of (a1, b1) and a2,b2 to clarify the impact of prior PDFs on the estimates of E-Bayesian method for λ and θ. The prior PDFs are given by(27)π1aj,bj=2ajcj,0<aj<1,0<bj<cj,π2aj,bj=2bjcj2,0<aj<1,0<bj<cj,π3aj,bj=3bj2cj3,0<aj<1,0<bj<cj,j=1,2.The E-Bayesian estimated value ofλ is obtained using the SEL function from (19), (25), and (27) as follows:(28)λ^EBSi=∬λ^BSa1,b1πia1,b1da1db1,i=1,2,3.Additionally, the E-Bayesian estimate ofλ may be produced through using LINEX loss function from (21), (25), and (27) as follows:(29)λ^EBLi=∬λ^BLa1,b1πia1,b1da1db1,i=1,2,3.


## 5.2. E-Bayesian Estimate ofθ

Based on the SEL and LINEX loss functions, the E-Bayesian estimate is obtained by using (27).We can use (20), (26), and (27) to obtain the estimates of θ as follows:(30)θ^EBSi=∬θ^BSπia1,b1da2db2,i=1,2,3,and for the LINEX loss function, we can use (22), (26), and (27) to obtain the estimate of the same parameter:(31)θ^EBLi=∬θ^BLπia1,b1da2db2,i=1,2,3.We employ the MCMC methodology to generate Bayesian and E-Bayesian estimates ofλ and θ because all estimators of Bayesian and E-Bayesian methods cannot be expressed explicitly as shown in the preceding equations.


## 6. MCMC Technique

We apply the MCMC technique when we fail to compute the estimates for the parametersλ and θ mathematically. Therefore, we turn to one of the most efficient methods of generating samples from a certain distribution, which is the Metropolis–Hastings method, and subsequently obtain Bayesian estimates. We make effective use of the Metropolis–Hastings approach, which generates randomly selected samples from any proposal distribution. The following are the posterior conditional PDFs of λ and θ, respectively.(32)π∗λ|θ=λr+p+a1−1e−b1λ1−1−e−s2λn1−r×∏j=1r1−e−tj2λ−11−1−e−θd2λn2−p×∏i=1p1−e−θxi2λ−1,(33)π∗θ|λ=θ2p+a2−1e−b2θ1−1−e−θd2λn2−p×∏i=1pe−θxi21−e−θxi2λ−1.Now we encounter a difficulty: we are unable to produce samples from the conditional posterior PDFs ofλ and θ, since it is self-evident that they cannot be simplified to a well-known model as shown in equations (32) and (33), and thus we adopt a normal distribution as a proposal distribution.Algorithm 1: Steps of algorithm.
Step 1: selectλ and θ as beginning values, say λ^MLE,θ^MLE.Step 2: start the counter with countj=1.Step 3: create a random sample ofλj from (32) using normal distribution.Step 4: compute the rejection-acceptance condition:rλj−1|λ∗=min1,π∗λ∗|θj−1/π∗λj−1|θj−1Step 5: we want to generate a random number that varies and ranges from 0 to 1, so we will turn to the uniform distributionU∼U0,1.Step 6: ifU≤rλj−1|λ∗, accept the proposal and set λ∗=λj; otherwise, reject the proposal and set λj−1=λj.Step 7: using the Metropolis–Hastings method, generateθj from (33) using normal distribution.Step 8: in order to find the estimate ofθ, we should make a repetition for steps 4–6.Step 9: make an increment in the loop counter likej=j+1.Step 10: repeat steps 3–9 for a total of N times.Step 11: under the SEL function, the Bayesian estimates regardingλ and θ have been provided by the following:λ^BS=1/N−M∑j=M+1Nλjθ^BS=1/N−M∑j=M+1NθjwhereM is a burn-in period that can be chosen.Step 12: the estimates for the two parametersλ and θ under the LINEX loss function can be easily computed, respectively, using the following equations:λ^BL=−1/hln1/N−M∑j=M+1Ne−hλj


## 7. Simulation Study

As per the following algorithm, a simulation study is conducted. All calculations are carried out using Mathematica 12 programming language software.(i)
Determine the values ofn,τ,h,a1,a2,b1,b2,c1,c2.(ii)
We start with chosen initial values for both parametersλ and θ.(iii)
We want to generate a random number that varies and also ranges from 0 to 1, so we will turn to the uniform distributionU∼U0,1.(iv)
Create a generalized Rayleigh Type-II hybrid censoring sample using the CSPALT framework and the inverse function technique as described in the following: a normal use condition case is given byX=−ln1−U1/λ1/2, and an accelerated use condition case takes the following form:(34)T=1θ−ln1−U1/λ1/2.(v)
Construction of a Markov chain comprising 11000 values ofλ and θ is accomplished via the use of the Metropolis–Hastings method, with the first 1000 discarded as “burn-in.”(vi)
We construct estimated values of Bayesian estimatesλ^BS and θ^BS under SEL function from (35) and (36), respectively, using MCMC samples.(vii)
Using (37) and (38), we develop Bayesian estimatesλ^BL and θ^BL, respectively, under the LINEX loss function.(viii)
We use the SEL function to construct E-Bayesian estimatesλ^EBS and θ^EBS, respectively, from (28) and (30).(ix)
From (29) and (31), E-Bayesian estimates λ^EBL and θ^EBL are, respectively, obtained using LINEX loss function.(x)
We computeλ^−λ2 where λ^ is the estimate of λ.(xi)
We computeθ^−θ2 where θ^ is the estimate of θ.(xii)
We compute mean squared error (MSE) of estimatesλ and θ, respectively, by(35)MSEλ^=11000∑λ^−λ2,MSEθ^=11000∑θ^−θ2.(xiii)
Tables1 and 2 are used to present the numerical results.Table 1
Mean and MSE of Bayesian and E-Bayesian estimates forλ when λ=0.63542,θ=1.6342,c1=c2=1,a1=a2=0.76243,b1=b2=0.8645,andh=1.5.
nn1,n2r1,r2τCriteriaBayesian estimatesE-Bayesian estimatesλ^BSλ^BLλ^EBS1λ^EBS2λ^EBS3λ^EBL1λ^EBL2λ^EBL330 (12, 18)(10, 15)0.8Mean0.157590.156990.1583910.1795960.2328910.1577810.1788120.231578MSE0.230090.230650.2293480.2100790.1659030.2299150.2107710.1669121.5Mean0.182130.181170.1830540.207560.2691530.1820820.2063120.267063MSE0.206850.207740.2060310.1848580.1371680.2069240.1859460.13874445 (18, 27)(15, 22)0.8Mean0.176490.176120.1773890.2011360.2608240.1770150.2006560.260016MSE0.213280.213610.2124910.1920720.1461570.2128210.192470.1467211.5Mean0.254090.253210.2553740.2895620.375490.2544870.2884270.373602MSE0.149650.150370.1487140.1487140.0768150.1494430.1259830.07796560 (24, 36)(19, 29)0.8Mean0.313060.312450.3146460.3567690.4626410.3140280.3559770.461322MSE0.106140.106540.105140.0805310.0347040.1055470.0809870.0351931.5Mean0.317750.317360.3193570.3621110.4695670.3189620.3616030.468714MSE0.102070.102330.1010660.0762020.0300370.1013210.0764880.030337Table 2
Mean and MSEs of Bayesian and E-Bayesian estimates forθ when λ=0.63542,θ=1.6342,c1=c2=1,a1=a2=0.76243,b1=b2=0.8645,andh=1.5.
nn1,n2r1,r2τCriteriaBayesian estimatesE-Bayesian estimatesθ^BSθ^BLθ^EBS1θ^EBS2θ^EBS3θ^EBL1θ^EBL2θ^EBL330 (12, 18)(10, 15)0.8Mean1.01441.014191.019541.156031.499081.019321.155751.49862MSE0.385550.385760.3792190.2304620.0213110.3794310.2306480.0212661.5Mean1.02371.022821.028891.166631.512831.0281.165491.51094MSE0.374150.375070.3678580.2204940.0178820.368780.2213320.01785945 (18, 27)(15, 22)0.8Mean1.012551.012281.017681.153921.496341.017411.153581.49579MSE0.386980.387270.3806370.2313590.0201610.3809270.2316320.0202081.5Mean1.035371.034731.040611.179931.530071.039971.17911.52871MSE0.359660.360390.3534170.2077440.0131620.3541420.2084330.01331860 (24, 36)(19, 29)0.8Mean1.057681.056941.063041.205351.563041.062291.20441.56146MSE0.337030.33770.3309240.189950.0152160.3315990.1905130.0149041.5Mean1.098961.097691.104521.252391.624041.103251.250761.62131MSE0.296680.297810.2908540.1590170.0223650.2919790.1599370.021717


## 8. Data Analysis

This section contains a numerical example that observes the efficiency of the methods used in the paper and assesses how they operate using real data and not simulated data. Many authors used these data [25]. For seven different values of mole fraction, they used it as minority electron mobility for p-type Ga1−xAlxAs. Tarvirdizade and Filliben [26] supplied just two sets of data for the mole fractions 0.25 and 0.30, claiming that the generalized Rayleigh distribution is well fitted for datasets 1 and 2 individually. Each dataset comprises 21 observations, as seen below (Table3).Table 3
Real data sets 1 and 2.
Dataset 1 (0.25)3.0512.7792.6042.3712.2142.0451.7151.5251.2961.1541.0160.79480.70070.62920.61750.64490.88811.1151.3971.5061.528Dataset 2 (0.30)2.6582.4342.2882.0921.9591.8141.5301.3661.1651.0410.91980.72410.64030.5760.56470.58730.80131.0021.2501.3471.368Here, CSPALT with Type-II HCS was applied on these uncensored data, withn1=10 and n2=11 units chosen at random from a total of n=21. We see the following for r1=9, r2=10, τ=2, λ=0.73625, and θ=1.6325.Dataset 1:(i)
Whenτ=2, the test is terminated at random if the usage conditions are normal. T∗=maxTr1,τ=max2.371,2=2.371, and hence 9 items will fail from 10 items. In the scenario of accelerated use, we find that the random time T∗=maxXr2,τ=max1.69949,2=2; as a result, out of 11 items, 11 items fail.Dataset 2:(i)
Whenτ=2, the test is ended at random time T∗=maxTr1,τ=max2.288,2=2.288; then, 9 items fail out of 10 items in the typical use condition. We can see that the random time T∗=maxXr2,τ=max0.836595,2=2 regarding accelerated case, resulting in 11 items failing out of 11 units.The same prior approaches are used to construct Bayesian and E-Bayesian estimates ofλ and θ under the SEL and LINEX loss functions for real datasets. Table 4 gives the numerical results for both datasets.Table 4
Mean and MSE of Bayesian and E-Bayesian methods forλ and θ when θ=1.6526,λ=0.7542,c1=c2=2,a1=a2=0.85243,b1=b2=1.6243, and h=1.5.
Estimated values ofλ for dataset 1n1,n2r1,r2τCriteriaBayesian estimatesE-Bayesian estimatesλ^BSλ^BLλ^EBS1λ^EBS2λ^EBS3λ^EBL1λ^EBL2λ^EBL321 (10, 11)(9, 10)2Mean0.309 870.306 720.365 7350.348 4530.424 4940.361 3620.344 480.418 621MSE0.198 110.200 90.151 8630.165 5010.109 9970.155 2490.168 7130.113 85721 (10, 11)(9, 10)2Estimates ofθ for dataset 1θ^BSθ^BLθ^EBS1θ^EBS2θ^EBS3θ^EBL1θ^EBL2θ^EBL3Mean1.012 371.012 251.194 871.138 411.386 841.194 711.138 261.386 62MSE0.387 920.388 070.193 8890.246 7990.061 6830.194 0370.246 9520.061 79621 (10, 11)(9, 10)2Estimates ofλ for dataset 2λ^BSλ^BLλ^EBS1λ^EBS2λ^EBS3λ^EBL1λ^EBL2λ^EBL3Mean0.231 110.227 170.272 7720.259 8830.316 5960.267 3430.254 9390.309 359MSE0.275 440.279 480.234 3040.246 6470.194 9070.239 4170.251 4330.201 07821 (10, 11)(9, 10)2Estimates ofθ for dataset 2θ^BSθ^BLθ^EBS1θ^EBS2θ^EBS3θ^EBL1θ^EBL2θ^EBL3Mean1.026 021.025 661.210 991.153 761.405 541.210 491.153 311.404 88MSE0.371 420.371 850.180 4040.232 1870.053 3440.180 8170.232 6140.053 638


## 9. Concluding Remarks

We estimated the distribution parameterλ by many methods such as classical Bayesian and E-Bayesian estimation methods and the acceleration factor θ of the CSPALT model from generalized Rayleigh distribution under Type-II HCS. It is noted that Bayesian estimators cannot be stated in a closed form; therefore, we utilize the MCMC technique to obtain estimates of the parameter λ and the acceleration factor θ using SEL and LINEX loss functions. In terms of MSE, the E-Bayesian estimation approach under SEL and LINEX loss functions performs better than the Bayesian estimation technique under the same loss functions for λ and θ, as shown in Tables 1 and 2. When sample size n and censoring time τ increase, the MSE of both Bayesian and E-Bayesian estimation methods decreases. Furthermore, as shown in Table 4, all previous findings are confirmed when actual datasets (1, 2) are used, demonstrating that the suggested procedures are effective in practice. Moreover, in practice, the E-Bayesian methodology outperforms the Bayesian method. At the end, because of the importance of the competing risks model in medical studies and reliability analysis, we can suggest “Bayesian inference in a constant-stress partially accelerated competing risks model based on hybrid censoring schemes” as future work.



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*Source: 6307435-2022-04-26.xml*