# Balanced Joint Progressively Hybrid Type-I Censoring Samples in Estimating the Lifetime Lomax Distributions

Authors: Nada M. Alfaer; Hassan M. Aljohani
Journal: Complexity
 (2021)

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## Abstract

The comparative life testing for products from different production lines under joint censoring schemes has received some attention over the past few years. Mondal and Kundu recently used the balanced joint progressive type-II censoring scheme to discuss the comparative exponential and Weibull populations. This paper implements the balanced censoring scheme with a hybrid progressive type-I censoring scheme known as a balanced joint progressive hybrid type-I censoring scheme (BJPHCS). The life Lomax products’ model formulation from two different lines of production with BJPHCS is discussed. The model parameters are estimated under maximum likelihood estimation for point and the corresponding asymptotic confidence intervals. Under independent gamma priors, the Bayes estimators and associated credible intervals are obtained with the help of MCMC technique. The validity of the theoretical results developed in this paper for estimation problems is discussed through numerical example and Monte Carlo simulation study, which report the estimators’ quality. Finally, we give a brief comment describing the numerical results.


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## Body

## 1. Introduction

The quality of any life product has required putting some product units under a life testing experiment, and the life data may be complete or censoring. The test cost and time determine a suitable method that is used to obtain the data. Type-I and type-II censoring schemes are presented as the oldest censoring schemes. The experiment is run to prefixed time in type-I censoring scheme, and the number of failed units is random. However, the experiment is run to prefixed number of failures in the Type-II censoring scheme with a random experiment time. In practice, if we need to remove units at any time of the experiment for engineering, clinical studies, or other purposes, then a progressive censoring scheme is applied to improve productivity while keeping a high level; see, for more details, [1–4]. The joint cases of type-I and type-II or progressive type-I and progressive type-II are called hybrid censoring schemes.Supposen independent units are randomly selected from the product to put under test and the priors, integer m, ideal test time τ, and censoring scheme R = {R1,R2, …, Rm} such as n=m+∑i=1mRi are determined. At each failure time Ti, Ri survival units are removed from the test, where i=1, 2, …, m. In progressive hybrid type-I censoring scheme, the experiment is terminated at min (τ,Tm), but, in a progressive hybrid type-II censoring scheme, the experiment is terminated at max (τ,Tm).In particular, when the product comes from different production lines, a joint censoring scheme appears as a suitable censoring scheme, and the obtaining data are used to determine the relative merits of life products in competing duration. Therefore, suppose two lines of productionA1 and A2 produced the same product under the same conditions. Suppose two independent samples κ1 and κ1 are selected from the lines A1 and A2, respectively, to be put simultaneously under life testing experiment. In the literature, joint censoring has been discussed in [5, 6]. The classical estimation is discussed in [7, 8]. Bayes estimation is presented by [9] and recently by [10–12].For jointly progressive hybrid type-I censoring scheme, the total sample with size (κ1+κ2 which is collected from the lines A1 and A2, where κ1 from A1 and κ2 from A2, are put under the test. Then, the number of failure units m, the censoring scheme R¯=R1,R2,…,Rm, and the ideal test time τ is given at the prior of the experiment. The failure time and the corresponding unit type (from A1 or A2), say (Ti, wi), is record for i=1,2,…,m, where wi=1, 0 is denoted to unit type, from line A1 or line A2, respectively. Then, at any step of the experiment, when the failure time (Ti, wi) is observed, Ri survival units are randomly removed from the experiment. The experiment is terminated when min τ,Tm is recorded and the random sample, say T=T1,w1,T2,w2,…,TJ,wJ,1≤J≤m, is obtained. However, if we consider the jointly progressive hybrid type-II censoring scheme, the experiment is terminated when the max (τ, Tm) is observed and the random sample, say T=T1,w1,T2,w2,…,TJ,wJ,m≤J≤n, is obtained.Balanced joint progressive type-II censoring is introduced early by [13] which is easier to handle than the joint progressive type-II censoring scheme. Different statistical properties of two exponential distributions under this scheme are discussed by [13]. This study is extended for Weibull lifetime distribution by [14]. Recently, inferences of Weibull parameters are underbalance two-sample type-II progressive censoring scheme [15]. In a balanced joint progressive type-II censoring scheme, we suppose two lines of production, A1 and A2, have the same kind of products under the same facility and the sample of size κ1+κ2 with size κ1 from A1 and κ2 from A2 are put under life testing. Also, the prior integer m is denoted to the number of failed units, and censoring scheme R¯=R1,,R2,…,Rm−1 is determined to satisfy m+∑i=1m−1Ri<minκ1,κ2. If the first failure T1 is observed from the line A1, then R1 and R1+1 survival units are removed from the sample κ1 and κ2, respectively. Also, if the second failure T2 is observed from the line A2, then R2+1 and R2 survival units are removed from the sample κ1−1−R1 and κ2−1−R1, respectively. Then, the test is continual until mth failure is reached. If the mth failure is from the line A1, then the remaining κ1−m−∑i=1m−1Ri and κ2−m−1−∑i=1m−1Ri survival units drown from the test. And, if the mth failure is from the line A2, then the remaining κ1−m−1−∑i=1m−1Ri and κ2−m−∑i=1m−1Ri survival units drown from the test. Then, the observed balanced joint progressive type-II censoring sample is given by t¯=t1,w1,t2,w2,…,tm,wm.This paper aims to use the progressive hybrid type-I censoring scheme under a balanced joint technique to present the BJPHCS. The BJPHCS is analytically easier to handle than another joint censoring scheme. Also, the properties of the estimators under BJPHCS can be stated more explicitly. Under this scheme, we construct the likelihood function for the model parameters. The lifetime information obtained from Lomax lifetime distribution under BJPHCS is used to present the two Lomax life distributions’ statistical inference. Also, the asymptotic confidence interval under the normality distribution of the estimates is constructed. The Bayes estimators of unknown model parameters with the corresponding credible intervals are developed. Different estimators are discussed and compared through the Monte Carlo simulation study and illustrated through a numerical example based on BJPHCS.The paper is constructed with the following sections. The concepts and model of the formulation are built in Section2. The model parameters are estimated with maximum likelihood estimation in Section 3. The Bayes estimation and the corresponding credible intervals are derived in Section 4. Numerical discussion in the form of illustrative example and Monto Carlo simulation studies are reported in Section 5. Some comments are built to discuss the numerical results in Section 6.


## 2. The Model

Suppose the product comes from different two linesA1 and A2 of production under the same conditions. And, let the sample of size κ1+κ2 be randomly selected with κ1 drawn from the line A1 and κ2 drawn from the line A2. Prior to the beginning of the experiment, the experimenter determines the number m of failure units, the ideal test time τ, and the nonnegative censoring scheme R¯=R1,R2,…,Rm to satisfy m+∑i=1m−1Ri<minκ1,κ2. When the experiment is beginning, the time to failure and its type are recorded, which means from line A1 or line A2. If the first failure T1 is recorded from the line A1, then R1 and R1+1 survival units are removed from the sample κ1 and κ2, respectively. Also, if the second failure T2 is observed from the line A2, then R2+1 and R2 survival units are removed from the sample κ1−R1−1 and κ2−R1−1, respectively. Then, the experiment is continual until the min τ,Tm is observed. If Tm<τ, the results are similar to balanced joint progressive type-II censoring [13], but, if Tm>τ, the experiment is removed at τ, where TJ<τ<TJ+1. If the Jth failure is from the line A1, then the remaining κ1−J−∑i=1J−1Ri and κ2−J−1−∑i=1J−1Ri survival units drown from the test. And, if the Jth failure is from the line A2, then the remaining κ1−J−1−∑i=1J−1Ri and κ2−J−∑i=1J−1Ri survival units drown from the test. Then, the observed BJPHC sample is given by t¯=t1,w1,t2,w2,…,tJ,wJ. Figure 1 shows the plane of BJPHCS. Suppose κ1 units have the identical independent distributed (i.i.d.) random lifetimes X1,X2,…,Xκ1 and κ2 units have the i.i.d. random lifetimes Y1,Y2,…,Yκ2. The two samples are distributed with distribution with probability density functions (PDFs), and the cumulative distribution functions (CDFs) are given by fl. and Fl.,l=1,2. Then, the ordered sample T1,T2,…,TJ, 1≤J≤m, is formulated from the joint sample X1,X2,…,XJ1,Y1,Y2,…,YJ2, where J=J1+J2 and J1 is the number of units fails from the line A1 and J2 is the number of units fails from the line A2. The observed BJPHCS censoring sample t¯=t1,w1,t2,w2,…,tJ,wJ in which wi is 1 or 0 values and depends on line A1 or A2, respectively, and J1=∑i=1Jwi and J2=∑i=1J1−wi.Figure 1
Schematic diagram of BJPHCS.
(a)(b)Hence, under observed samplet¯=t1,w1,t2,w2,…,tJ,wJ, the likelihood function is given by(1)f1,2,…,Jt¯|Φ∝∏i=1Jh1tiwih2ti1−wiS1tiS2tiRi+1×S1τ∗κ1−J−∑i=1JRiS2τ∗κ2−J−∑i=1JRi,where τ∗=minτ,Tm, Φ is the parameters vector, and Sj. and hj., j=1,2, are denoted as reliability and hazard rate functions, respectively. Different lines A1 and A2 of production with units have PDFs, and CDFs follow Lomax lifetime distribution defined with PDFs and given by(2)flt=αlβl1+tβl−αl+1,t>0,αl,βl>0,j=1,2,and CDFs are given by(3)Flt=1−1+tβl−αl.The reliability and hazard rate functions of Lomax lifetime distributions is also given by(4)Slt=1+tβl−αl,(5)hlt=αlβl+t.The Lomax lifetime distribution has been introduced early in [16] and more information about this distribution has been introduced in [17]. Lomax lifetime distribution is heavy-tailed; this propriety makes a more suitable distribution than exponential, gamma, and Weibull lifetime distribution. The failure time of Lomax lifetime distribution under different risks is studied by [18] and recently by [19].


## 3. Maximum Likelihood Estimation

For the given joint samplet¯={t1,w1, t2,w2, …, tJ,wJ} and Lomax distribution given by (2)–(5), the joint likelihood function (1) is reduced to(6)Lα1,α2,β1,β2|t¯∝α1J1α2J2∏i=1Jβ1+ti−wiβ2+ti−1−wi1+tiβ1−α1Ri+11+tiβ2−α2Ri+11+τ∗β1−α1K11+τ∗β2−α2K2,where(7)Ki=κi−J−∑i=1JRi,i=1,2.The joint likelihood function is defined by (6) after the natural logarithms are reduced to(8)ℓα1,α2,β1,β2|t¯=J1logα1+J2logα2−∑i=1Jwilogβ1+ti−∑i=1J1−wi×logβ2+ti−∑i=1JRi+1α1log1+tiβ1+α2log1+tiβ2−α1K1log1+τ∗β1−α2K2log1+τ∗β2.The likelihood equations can be obtained from logarithm (8) by taking the first partial derivative with respect to model parameters, and hence, the point and interval estimate can be formulated as follows.


### 3.1. MLEs

LetΦ=α1,α2,β1,β2 be the model parameters vector; then, the zero values of derivative of log-likelihood function are described by,(9)∂ℓα1,α2,β1,β2|t¯∂αj=0,j=1,2,and described by,(10)α1=J1∑i=1JRi+1log1+ti/β1+K1log1+τ∗/β1,(11)α2=J2∑i=1JRi+1log1+ti/β2+K2log1+τ∗/β2.Also,(12)∂ℓα1,α2,β1,β2|t¯∂βj=0,j=1,2,which is described by,(13)−∑i=1Jwiβ1+ti+α1β1∑i=1JRi+1tiβ1+ti+K1τ∗β1+τ=0,(14)−∑i=1J1−wiβ2+ti+α2β2∑i=1JRi+1tiβ2+ti+K2τ∗β2+τ=0.Equations (13) and (14) after replacing αi, i = 1, 2, from (10) and (11), are reduced to(15)−∑i=1Jwiβ1+ti+J1/β1∑i=1JRi+1ti/β1+ti+K1τ∗/β1+τ∑i=1JRi+1log1+ti/β1+K1log1+τ∗/β1=0,(16)−∑i=1J1−wiβ2+ti+J/β2∑i=1JRi+1ti/β2+ti+K2τ∗/β2+τ∑i=1JRi+1log1+ti/β2+K2log1+τ∗/β2=0.The nonlinear equations (15) and (16) are simply solved with Newton–Raphson iteration to obtain β^1 and β^2. Then, the estimates α^1 and α^2 are obtained from (10) and (11).Remark 1.
For each of the parameters,αi and βi, i=1, 2, do not exist for the case J1=0 as well as J2=0. Also, if J1=1 as well as J2=1, the parameters αi and βi are not estimable. Hence, the MLEs in (10), (11), (15), and (16) are only conditional MLEs, conditioned on 1<J1and 1<J2. Hence, the properties of the MLEs are discussed only as conditional on 1<J1and 1<J2 [20]. Also, the exact distributions for estimators Φ^=α^1,α^2,β^1,β^2 are difficult to obtain [21].


### 3.2. Approximate Confidence Interval

Fisher information matrix is defined as the minus expectation of the second derivative of log-likelihood function with respect to the model parameters’ vectorsΣ=−E∂2ℓα1,α2,β1,β2|t¯/∂Φi∂Φj, i, j=1, 2, 3, 4. For the cases of a large sample, we can use the approximate information matrix, which is obtained under maximum likelihood estimates and denoted by Σ0α^1,α^2,β^1,β^2 which is applied as alternative of Fisher information matrix. The approximate confidence intervals for model parameters Φ=α1,α2,β1,β2 are dependent on the asymptotic normality distribution of estimates α^1,α^2,β^1, and β^2 with mean α1,α2,β1,β2 and variance covariance matrix Σ0−1α^1,α^2,β^1,β^2; then, we can say that Φ^=α^1,α^2,β^1,β^2 is distributed as normal distribution as follows:(17)Φ^∼Nα1,α2,β1,β2,Σ0−1α^1,α^2,β^1,β^2,and Σ0−1α^1,α^2,β^1,β^2 is given by(18)Σ0−1α^1,α^2,β^1,β^2=−∂2ℓα1,α2,β1,β2|t¯∂α12−∂2ℓα1,α2,β1,β2|t¯∂α1∂α2−∂2ℓα1,α2,β1,β2|t¯∂α1∂β1−∂2ℓα1,α2,β1,β2|t¯∂α1∂β2−∂2ℓα1,α2,β1,β2|t¯∂α2∂α1−∂2ℓα1,α2,β1,β2|t¯∂α22−∂2ℓα1,α2,β1,β2|t¯∂α2∂β1−∂2ℓα1,α2,β1,β2|t¯∂α2∂β2−∂2ℓα1,α2,β1,β2|t¯∂β1∂α1−∂2ℓα1,α2,β1,β2|t¯∂β1∂α2−∂2ℓα1,α2,β1,β2|t¯∂β12−∂2ℓα1,α2,β1,β2|t¯∂β1∂β2−∂2ℓα1,α2,β1,β2|t¯∂β2∂α1−∂2ℓα1,α2,β1,β2|t¯∂β2∂α2−∂2ℓα1,α2,β1,β2|t¯∂β2∂β−∂2ℓα1,α2,β1,β2|t¯∂β22,at Φ^=α^1,β^1,α^2,β^2.Then, at confidence levelγ, 1001−γ% approximate confidence intervals of α1,α2,β1, and β2, respectively, are given by(19)α^i∓zγ/2eiiβ^i∓zγ/2ei+2i+2,i=1,2,where the values e11,e22,e33, and e44 are the diagonal of matrix Σ0−1 with standard normal value zγ/2 with right-tail probability γ/2. The second derivative of log-likelihood function with respect to the model parameters vector Φ=α1,α2,β1,β2 are given by(20)∂2ℓα1,α2,β1,β2|t¯∂αi2=−Jiαi2,i=1,2,∂2ℓα1,α2,β1,β2|t¯∂β12=∑i=1Jwiβ1+ti2−α1β12∑i=1JRi+1ti2β1+tiβ1+ti2−K1τ2β1+τ∗β1+τ2,∂2ℓα1,α2,β1,β2|t¯∂β22=∑i=1J1−wiβ2+ti2−α2β22∑i=1JRi+1ti2β2+tiβ2+ti2−K2τ2β2+τβ2+τ2,∂22ℓα1,α2,β1,β2|t¯∂α1∂β1=∂22ℓα1,α2,β1,β2|t¯∂β1∂α1=1β1∑i=1JRi+1tiβ1+ti+K1τ∗β1+τ,∂2ℓα1,α2,β1,β2|t¯∂α2∂β2=∂2ℓα1,α2,β1,β2|t¯∂β2∂α2=1β2∑i=1JRi+1tiβ2+ti+K2τ∗β2+τ,∂2ℓα1,α2,β1,β2|t¯∂α1∂β2=∂2ℓα1,α2,β1,β2|t¯∂β2∂α1=∂2ℓα1,α2,β1,β2|t¯∂α1∂α2=∂2ℓα1,α2,β1,β2|t¯∂α2∂α1=∂2ℓα1,α2,β1,β2|t¯∂α2∂β1=∂2ℓα1,α2,β1,β2|t¯∂β1∂α2=∂2ℓα1,α2,β1,β2|t¯∂β1∂β2=∂2ℓα1,α2,β1,β2|t¯∂β2∂β1=0.


## 3.1. MLEs

LetΦ=α1,α2,β1,β2 be the model parameters vector; then, the zero values of derivative of log-likelihood function are described by,(9)∂ℓα1,α2,β1,β2|t¯∂αj=0,j=1,2,and described by,(10)α1=J1∑i=1JRi+1log1+ti/β1+K1log1+τ∗/β1,(11)α2=J2∑i=1JRi+1log1+ti/β2+K2log1+τ∗/β2.Also,(12)∂ℓα1,α2,β1,β2|t¯∂βj=0,j=1,2,which is described by,(13)−∑i=1Jwiβ1+ti+α1β1∑i=1JRi+1tiβ1+ti+K1τ∗β1+τ=0,(14)−∑i=1J1−wiβ2+ti+α2β2∑i=1JRi+1tiβ2+ti+K2τ∗β2+τ=0.Equations (13) and (14) after replacing αi, i = 1, 2, from (10) and (11), are reduced to(15)−∑i=1Jwiβ1+ti+J1/β1∑i=1JRi+1ti/β1+ti+K1τ∗/β1+τ∑i=1JRi+1log1+ti/β1+K1log1+τ∗/β1=0,(16)−∑i=1J1−wiβ2+ti+J/β2∑i=1JRi+1ti/β2+ti+K2τ∗/β2+τ∑i=1JRi+1log1+ti/β2+K2log1+τ∗/β2=0.The nonlinear equations (15) and (16) are simply solved with Newton–Raphson iteration to obtain β^1 and β^2. Then, the estimates α^1 and α^2 are obtained from (10) and (11).Remark 1.
For each of the parameters,αi and βi, i=1, 2, do not exist for the case J1=0 as well as J2=0. Also, if J1=1 as well as J2=1, the parameters αi and βi are not estimable. Hence, the MLEs in (10), (11), (15), and (16) are only conditional MLEs, conditioned on 1<J1and 1<J2. Hence, the properties of the MLEs are discussed only as conditional on 1<J1and 1<J2 [20]. Also, the exact distributions for estimators Φ^=α^1,α^2,β^1,β^2 are difficult to obtain [21].


## 3.2. Approximate Confidence Interval

Fisher information matrix is defined as the minus expectation of the second derivative of log-likelihood function with respect to the model parameters’ vectorsΣ=−E∂2ℓα1,α2,β1,β2|t¯/∂Φi∂Φj, i, j=1, 2, 3, 4. For the cases of a large sample, we can use the approximate information matrix, which is obtained under maximum likelihood estimates and denoted by Σ0α^1,α^2,β^1,β^2 which is applied as alternative of Fisher information matrix. The approximate confidence intervals for model parameters Φ=α1,α2,β1,β2 are dependent on the asymptotic normality distribution of estimates α^1,α^2,β^1, and β^2 with mean α1,α2,β1,β2 and variance covariance matrix Σ0−1α^1,α^2,β^1,β^2; then, we can say that Φ^=α^1,α^2,β^1,β^2 is distributed as normal distribution as follows:(17)Φ^∼Nα1,α2,β1,β2,Σ0−1α^1,α^2,β^1,β^2,and Σ0−1α^1,α^2,β^1,β^2 is given by(18)Σ0−1α^1,α^2,β^1,β^2=−∂2ℓα1,α2,β1,β2|t¯∂α12−∂2ℓα1,α2,β1,β2|t¯∂α1∂α2−∂2ℓα1,α2,β1,β2|t¯∂α1∂β1−∂2ℓα1,α2,β1,β2|t¯∂α1∂β2−∂2ℓα1,α2,β1,β2|t¯∂α2∂α1−∂2ℓα1,α2,β1,β2|t¯∂α22−∂2ℓα1,α2,β1,β2|t¯∂α2∂β1−∂2ℓα1,α2,β1,β2|t¯∂α2∂β2−∂2ℓα1,α2,β1,β2|t¯∂β1∂α1−∂2ℓα1,α2,β1,β2|t¯∂β1∂α2−∂2ℓα1,α2,β1,β2|t¯∂β12−∂2ℓα1,α2,β1,β2|t¯∂β1∂β2−∂2ℓα1,α2,β1,β2|t¯∂β2∂α1−∂2ℓα1,α2,β1,β2|t¯∂β2∂α2−∂2ℓα1,α2,β1,β2|t¯∂β2∂β−∂2ℓα1,α2,β1,β2|t¯∂β22,at Φ^=α^1,β^1,α^2,β^2.Then, at confidence levelγ, 1001−γ% approximate confidence intervals of α1,α2,β1, and β2, respectively, are given by(19)α^i∓zγ/2eiiβ^i∓zγ/2ei+2i+2,i=1,2,where the values e11,e22,e33, and e44 are the diagonal of matrix Σ0−1 with standard normal value zγ/2 with right-tail probability γ/2. The second derivative of log-likelihood function with respect to the model parameters vector Φ=α1,α2,β1,β2 are given by(20)∂2ℓα1,α2,β1,β2|t¯∂αi2=−Jiαi2,i=1,2,∂2ℓα1,α2,β1,β2|t¯∂β12=∑i=1Jwiβ1+ti2−α1β12∑i=1JRi+1ti2β1+tiβ1+ti2−K1τ2β1+τ∗β1+τ2,∂2ℓα1,α2,β1,β2|t¯∂β22=∑i=1J1−wiβ2+ti2−α2β22∑i=1JRi+1ti2β2+tiβ2+ti2−K2τ2β2+τβ2+τ2,∂22ℓα1,α2,β1,β2|t¯∂α1∂β1=∂22ℓα1,α2,β1,β2|t¯∂β1∂α1=1β1∑i=1JRi+1tiβ1+ti+K1τ∗β1+τ,∂2ℓα1,α2,β1,β2|t¯∂α2∂β2=∂2ℓα1,α2,β1,β2|t¯∂β2∂α2=1β2∑i=1JRi+1tiβ2+ti+K2τ∗β2+τ,∂2ℓα1,α2,β1,β2|t¯∂α1∂β2=∂2ℓα1,α2,β1,β2|t¯∂β2∂α1=∂2ℓα1,α2,β1,β2|t¯∂α1∂α2=∂2ℓα1,α2,β1,β2|t¯∂α2∂α1=∂2ℓα1,α2,β1,β2|t¯∂α2∂β1=∂2ℓα1,α2,β1,β2|t¯∂β1∂α2=∂2ℓα1,α2,β1,β2|t¯∂β1∂β2=∂2ℓα1,α2,β1,β2|t¯∂β2∂β1=0.


## 4. Bayes Estimation

The Bayesian approach for parameters’ estimation depends on the prior information available about the parameters and the data presented. Gamma distribution characterized by different shapes depends on its parameters; this property has marked it to be a more suitable distribution in other cases. So, we consider the independent gamma prior for distribution of model parameters as follows:(21)Πl∗Φl=blalΓalΦlal−1exp−blΦl,l=1,2,3,4,where Φ presents the parameters’ vector. The joint prior density is given by(22)Π∗Φ=∏l=14Πl∗Φl.Also, after obtaining the data in the form of balanced joint progressive hybrid type-I censoring data which formed by likelihood function, then, we construct all information about the parameters by posterior probability density defined by(23)ΠΦ|t¯=QΠ∗ΦLΦ|t¯,where Q=∫ΦΠ∗ΦLΦ|t¯dΦ. Then, the Bayes estimators for the function ζΦ under squared error loss function (SEL) is given by(24)Φ^B=Q∫ΦζΦΠ∗ΦLΦ|t¯dΦ.Hence, the Bayes estimators have two integrables, which need approximation methods such as numerical integration or Lindley approximation. The more general case is applied in different Bayesian computation areas called as Markov chain Monto Carlo (MCMC) method.MCMC approach: to adopt the Bayesian approach under the MCMC method, we obtain the conditional distributions of the posterior distribution, which is given by(25)Πα1,α2,β1,β2|t¯∝exp−b1α1−b2α2−b3β1−b4β2−∑i=1Jwilogβ1+ti−∑i=1J1−wi×logβ2+ti−∑i=1JRi+1α1log1+tiβ1+α2log1+tiβ2−α1K1×log1+τ∗β1−α2K2log1+τ∗β2α1J1+a1−1α2J2+a2−1β1a3−1β2a4−1.The full conditional distributions of the parameters are given by(26)Πα1|α2,β1,β2t¯∝α1J1+a1−1exp−α1b1+∑i=1JRi+1log1+tiβ1+K1log1+τ∗β1,(27)Πα2|α1,β1,β2,t¯∝α1J2+a2−1exp−α2b2+∑i=1JRi+1log1+tiβ2+K2log1+τ∗β2,(28)Πβ1|α1,α2,β2,t¯∝β1a3−1exp−b3β1−∑i=1Jwilogβ1+ti−∑i=1JRi+1α1×log1+tiβ1−α1K1log1+τ∗β1,(29)Πβ2|α1,α2,β1,t¯∝β2a4−1exp−b4β2−∑i=1J1−wilogβ2+ti−α2∑i=1JRi+1×log1+tiβ2−α2K2log1+τ∗β2.Then, the posterior distribution is reduced to two conditional gamma functions and two distributions more similar to normal distribution. Then, the posterior distribution (25) and conditional posterior distributions (26) to (29) have shown that MCMC methods are more suitable with more general Metropolis–Hastings (MH) under Gibbs algorithms, see [22] described as follows.


### 4.1. MH under Gibbs algorithm

Step 1: for given initial vectorsΦ0=α^1,β^1,α^2,β^2, put I=1Step 2: generate two valuesα1I and α2I from conditional gamma distribution given by (26) and (27), respectivelyStep 3: with proposal normal distributions, generate two valuesβ1I and β2I from the conditional distributions (28) and (29)Step 4: construct the iteration vectorΦI=α1I,α2I,β1I,β2IStep 5: changeI to be I+1Step 6: steps (2) to (5) are followed to get iteration procedure toM timesStep 7: to reach the stationary state, we need iteration numberM∗ which is known by “burn-in;” then, the Bayes estimators is given by(30)ζ^ΦB=EΠ|t¯ζΦ|t¯=1M−M∗∑i=M∗+1MζiΦi,as well as the corresponding posterior variance ofζΦ also, which is given by(31)VζΦ|t¯=1M−M∗∑i=M∗+1MζiΦi−ζ^ΦB2,whereζΦ is a function of the parameters vector may be any one of them.Step 8: the obtaining empirical distribution ofζΦ is obtained by the MCMC iteration after arranging its values ascending; then, the corresponding 1001−γ% credible interval of ζΦ is given by(32)ζγ/2M−M∗,ζ1−γ/2M−M∗.


## 4.1. MH under Gibbs algorithm

Step 1: for given initial vectorsΦ0=α^1,β^1,α^2,β^2, put I=1Step 2: generate two valuesα1I and α2I from conditional gamma distribution given by (26) and (27), respectivelyStep 3: with proposal normal distributions, generate two valuesβ1I and β2I from the conditional distributions (28) and (29)Step 4: construct the iteration vectorΦI=α1I,α2I,β1I,β2IStep 5: changeI to be I+1Step 6: steps (2) to (5) are followed to get iteration procedure toM timesStep 7: to reach the stationary state, we need iteration numberM∗ which is known by “burn-in;” then, the Bayes estimators is given by(30)ζ^ΦB=EΠ|t¯ζΦ|t¯=1M−M∗∑i=M∗+1MζiΦi,as well as the corresponding posterior variance ofζΦ also, which is given by(31)VζΦ|t¯=1M−M∗∑i=M∗+1MζiΦi−ζ^ΦB2,whereζΦ is a function of the parameters vector may be any one of them.Step 8: the obtaining empirical distribution ofζΦ is obtained by the MCMC iteration after arranging its values ascending; then, the corresponding 1001−γ% credible interval of ζΦ is given by(32)ζγ/2M−M∗,ζ1−γ/2M−M∗.


## 5. Numerical Computations

### 5.1. Example

Discuss and illustrate the proposed methods in this paper, firstly, about the relation between choosing the prior information that related with true parameters’ values. The true parameters are chosen to satisfy thatEΦi≃ai/bi, see [23]. Also, the parameters are randomly generated from the known prior distribution; then, we can choose a set of parameters values generated from gamma distribution with parameters (a, b)={(2, 2), (4, 2), (3, 2), (5, 2)} to be Φ = {1.2, 2.2, 1.5, 2.5}. For the random selection, the two samples of size (κ1, κ2) = (40, 40) and prior integers m=25, censoring scheme R={14, 023, 1}, and the ideal test time τ=10. Hence, the randomly generated BJPHC data concerning to the algorithms are given by [24]. The generated data (ti,wi) under the last consideration is summarized in Table 1. For the data given in Table 1, the point MLE and corresponding confidence interval are summarized in Table 2. For the Bayesian approach, we run the chain 11,000 with the first 1000 value as burn-in the point, and the interval estimate is summarized in Table 2. Markov chain is built with desired properties and determines how many steps are needed to converge to the stationary distribution. The standard empirical method to assess convergence is also discussed with the plot of the first 11,000 steps for the Gibbs sampler shown in Figures 2 to 5. Then, the figures show the quality of the MCMC method in the Bayes method.Table 1
The generated balanced joint progressive type-I data.
0.030750.214310.232530.322290.399420.517260.801950.81130100100010.830830.863321.293081.301551.469591.486891.666292.03257110100102.162322.731662.753792.918974.333017.74818——110011——Table 2
Point and 95% confidence and credible intervals (ACIs and CIs) of MLEs Bayes estimates.
Pa.s(.)ML.BMCMC95% ACIsLength95% CIsLengthα1 = 1.20.72150.8097(0, 1.9320)1.9320(0.3969, 1.4132)1.01626β1 = 2.22.83682.8453(0, 3.9947)3.9947(1.4185, 4.8630)3.4445α2 = 1.50.44431.0680(0, 9.5860)9.5860(0.5572, 1.7875)1.23033β2 = 2.51.28653.2739(0, 3.7570)3.7570(1.8051, 5.2599)3.45486Figure 2
The simulation number and the corresponding histogram ofα1 generated by the MCMC method.
(a)(b)Figure 3
The simulation number and the corresponding histogram ofα2 generated by the MCMC method.
(a)(b)Figure 4
The simulation number and the corresponding histogram ofβ1 generated by the MCMC method.
(a)(b)Figure 5
The simulation number and the corresponding histogram ofβ2 generated by the MCMC method.
(a)(b)


### 5.2. Simulation Studies

This section has examined the best choice of sample size, censoring scheme, and ideal test time. This discussion is reported for the classical ML and Bayes estimators of Lomax lifetime distribution under a balanced joint progressive hybrid type-I censoring scheme. Hence, the developed theoretical results are compared and assessed with constructed Monte Carlo simulation studies. The point estimators are measured with mean estimate values (ME) and square rote of simulated variance (SV) which are given by(33)ME=Φ¯=1N∑i=1NΦi,SV=1N∑i=1NΦi−Φ¯2,where N is the iteration number of simulation and Φ is the parameters’ vector. In interval estimation, we adopted probability coverage (PC) and the mean interval length (MIL). In our simulation study, we consider the only set of the values of the parameters Φ = {3.0, 1.0, 2.0, 1.5}. A simulation is done with two values of τ=1.5,3.0 and two choices of prior information and noninformative prior (prior 0), in which posterior distribution is proportional with likelihood function and informative prior (prior 1) the values of prior parameters given by {(5.0, 1.0), (3.0, 2.5), (4.0, 2.0), (3, 2)}. The study is carried out under 1000 sets of balanced joint progressive type-I generated of Lomax lifetime. For each sample, the ML and Bayes estimate is reported, and the Bayesian approach is computed under SEL with 11,000 chains removed the first 1000 as burn-in.


## 5.1. Example

Discuss and illustrate the proposed methods in this paper, firstly, about the relation between choosing the prior information that related with true parameters’ values. The true parameters are chosen to satisfy thatEΦi≃ai/bi, see [23]. Also, the parameters are randomly generated from the known prior distribution; then, we can choose a set of parameters values generated from gamma distribution with parameters (a, b)={(2, 2), (4, 2), (3, 2), (5, 2)} to be Φ = {1.2, 2.2, 1.5, 2.5}. For the random selection, the two samples of size (κ1, κ2) = (40, 40) and prior integers m=25, censoring scheme R={14, 023, 1}, and the ideal test time τ=10. Hence, the randomly generated BJPHC data concerning to the algorithms are given by [24]. The generated data (ti,wi) under the last consideration is summarized in Table 1. For the data given in Table 1, the point MLE and corresponding confidence interval are summarized in Table 2. For the Bayesian approach, we run the chain 11,000 with the first 1000 value as burn-in the point, and the interval estimate is summarized in Table 2. Markov chain is built with desired properties and determines how many steps are needed to converge to the stationary distribution. The standard empirical method to assess convergence is also discussed with the plot of the first 11,000 steps for the Gibbs sampler shown in Figures 2 to 5. Then, the figures show the quality of the MCMC method in the Bayes method.Table 1
The generated balanced joint progressive type-I data.
0.030750.214310.232530.322290.399420.517260.801950.81130100100010.830830.863321.293081.301551.469591.486891.666292.03257110100102.162322.731662.753792.918974.333017.74818——110011——Table 2
Point and 95% confidence and credible intervals (ACIs and CIs) of MLEs Bayes estimates.
Pa.s(.)ML.BMCMC95% ACIsLength95% CIsLengthα1 = 1.20.72150.8097(0, 1.9320)1.9320(0.3969, 1.4132)1.01626β1 = 2.22.83682.8453(0, 3.9947)3.9947(1.4185, 4.8630)3.4445α2 = 1.50.44431.0680(0, 9.5860)9.5860(0.5572, 1.7875)1.23033β2 = 2.51.28653.2739(0, 3.7570)3.7570(1.8051, 5.2599)3.45486Figure 2
The simulation number and the corresponding histogram ofα1 generated by the MCMC method.
(a)(b)Figure 3
The simulation number and the corresponding histogram ofα2 generated by the MCMC method.
(a)(b)Figure 4
The simulation number and the corresponding histogram ofβ1 generated by the MCMC method.
(a)(b)Figure 5
The simulation number and the corresponding histogram ofβ2 generated by the MCMC method.
(a)(b)


## 5.2. Simulation Studies

This section has examined the best choice of sample size, censoring scheme, and ideal test time. This discussion is reported for the classical ML and Bayes estimators of Lomax lifetime distribution under a balanced joint progressive hybrid type-I censoring scheme. Hence, the developed theoretical results are compared and assessed with constructed Monte Carlo simulation studies. The point estimators are measured with mean estimate values (ME) and square rote of simulated variance (SV) which are given by(33)ME=Φ¯=1N∑i=1NΦi,SV=1N∑i=1NΦi−Φ¯2,where N is the iteration number of simulation and Φ is the parameters’ vector. In interval estimation, we adopted probability coverage (PC) and the mean interval length (MIL). In our simulation study, we consider the only set of the values of the parameters Φ = {3.0, 1.0, 2.0, 1.5}. A simulation is done with two values of τ=1.5,3.0 and two choices of prior information and noninformative prior (prior 0), in which posterior distribution is proportional with likelihood function and informative prior (prior 1) the values of prior parameters given by {(5.0, 1.0), (3.0, 2.5), (4.0, 2.0), (3, 2)}. The study is carried out under 1000 sets of balanced joint progressive type-I generated of Lomax lifetime. For each sample, the ML and Bayes estimate is reported, and the Bayesian approach is computed under SEL with 11,000 chains removed the first 1000 as burn-in.


## 6. Concluding Remarks

The commonly used problem in life products with different production lines is measuring the relative merits of the products competing for the duration. The main aim is to use censoring that can be applied to obtain the information in determining time. Recently, a joint censoring scheme and a specially balanced joint censoring scheme can be applied for this problem. BJPHC is proposed for obtaining information about lifetime Lomax products. In this paper, we consider the MLEs of unknown model parameters and the corresponding approximate confidence intervals. Also, we consider Baye’s estimation of the unknown parameters under consideration of the gamma priors on the unknown parameters. The Bayes estimates are computed with squared error loss functions. The Bayes estimators’ explicit forms cannot be obtained, so the MCMC approach is impalement to get the parameter estimation. The results are compared by using Monte Carlo simulation between the performances of the MLEs and approximate Bayes estimates under the assumptions of noninformative and informative priors. The Bayes estimates under the MCMC technique have been required to generate from the posterior distribution. The simulated MCMC sample plot and the corresponding histograms match quite well with the theoretical posterior density functions. The results can be extended to different types of loss functions. The proposed model and the results can also be easily extended for several members of the exponential family. Finally, we should mention that the products in Tables3 and 4 simulation study show that(1)
The life experiments under balanced jointly progressive type-I censoring scheme present a good source for obtaining the information about Lomax lifetime products(2)
The results improve for larger values of experiment total timeτ(3)
The Bayes estimation with informative prior distribution performs better than ML and noninformative prior method(4)
The results of MLE are closed to the Bayes estimates for noninformative prior(5)
For increase, the effective sample sizem estimators perform better in the form of SVs, interval length, and coverage probability(6)
Censoring scheme with middle censoring performs better than other censoring schemes(7)
The method of MCMC for approximation Bayesian estimate serves well, especially in dimensional casesTable 3
MEs andSVs of Lomax distributions with Φ= (3.0, 1.0, 2.0, 1.5).
τκ1,κ2(m,R)PaMLBMCMCprior0BMCMCprior1AVsMSEsAVsMSEsAVsMSEs1.5(35,30)(25,(5, 0(24)))α13.8321.2543.7101.1143.6540.984α21.4210.4261.3320.4201.3400.325β12.3150.7542.3740.7012.2140.654β21.9850.5131.8540.5001.8520.452(25,(0(10), 5, 0(10)))α13.7841.1243.7251.0123.6110.897α21.3890.4001.3250.3871.3510.301β12.3010.7152.3120.6982.2230.578β21.8520.4801.8110.4711.7770.399(45,45)(30,(15, 0(29)))α13.4520.9853.4110.9793.4120.745α21.3010.3451.3110.3411.3100.298β12.2980.6212.3130.6022.2050.495β21.8110.4071.8150.3991.7120.300(1.5, 30,(0(14), 15, 0(15)))α13.4150.9563.4150.9493.3640.711α21.2890.3231.3000.3101.2770.250β12.2500.5982.3130.5782.2110.428β21.7830.3851.8110.3701.7190.2873.0(30,35)(25,(5, 0(24)))α13.5000.9283.4900.9153.3310.697α21.2330.3011.2880.2991.2150.210β12.2180.5112.3100.5032.2000.401β21.7600.3221.7810.3151.7020.262(25,(0(10),5, 0(10)))α13.5120.9013.4950.8993.3120.656α21.2300.2981.2820.2811.2100.199β12.2140.4972.3010.4902.2130.395β21.7520.3031.7770.3071.7130.249(45,45)(30,(15, 0(29)))α13.5150.8893.4750.8523.3000.601α21.2170.2651.2250.2561.2120.187β12.2100.4602.2980.4552.2150.361β21.7440.2971.7780.2961.7100.215(30,(0(14),15, 0(15)))α13.4210.8603.4700.8443.2780.593α21.2100.2411.2110.2391.2100.161β12.2120.4482.2100.4372.2000.339β21.7370.2811.7550.2771.7120.203Table 4
MILs and PCs of Lomax distributions withΦ = (3.0, 1.0, 2.0, 1.5).
τκ1,κ2(m,R)PaMLBMCMCprior0BMCMCprior1MILsPCsMILsPCsMILsPCs1.5(35,30)(25,(5, 0))α17.235(0.77)7.027(0.88)6.548(0.89)α23.368(0.88)3.287(0.88)3.001(0.90)β15.214(0.89)5.147(0.89)4.512(0.89)β24.512(0.89)4.325(0.90)3.541(0.90)(25,(0(10),5, 0(10)))α17.122(0.89)6.920(0.90)6.442(0.90)α23.265(0.89)3.182(0.89)2.905(0.90)β15.118(0.91)5.100(0.91)4.470(0.89)β24.410(0.89)4.360(0.90)3.480(0.91)(45,45)(30,(15, 0(29)))α17.001(0.90)6.780(0.90)6.311(0.92)α23.170(0.89)3.165(0.92)2.798(0.92)β15.024(0.90)5.003(0.91)4.304(0.90)β24.312(0.90)4.299(0.91)3.320(0.91)(1.5, 30,(0(14),15, 0(15)))α16.902(0.91)6.700(0.91)6.140(0.93)α23.045(0.90)3.022(0.92)2.666(0.92)β14.920(0.89)4.901(0.93)4.123(0.94)β24.211(0.91)4.200(0.90)3.245(0.92)3.0(30,35)(25,(5, 0(24)))α16.450(0.90)6.320(0.90)5.824(0.90)α22.842(0.90)2.850(0.92)2.601(0.92)β14.754(0.90)4.701(0.91)4.013(0.93)β24.032(0.90)4.045(0.91)3.122(0.91)(25,(0(10),5, 0(10)))α16.361(0.91)6.311(0.92)5.715(0.94)α22.715(0.91)2.711(0.92)2.514(0.92)β14.620(0.92)4.601(0.92)3.992(0.93)β23.911(0.90)3.920(0.93)3.009(0.93)(45,45)(30,(15, 0(29)))α16.155(0.90)6.140(0.93)5.560(0.92)α22.511(0.94)2.531(0.92)2.347(0.92)β14.413(0.91)4.422(0.93)3.870(0.94)β23.741(0.90)3.700(0.91)2.891(0.91)(30,(0(14),15, 0(15)))α16.035(0.92)6.011(0.93)5.453(0.95)α22.415(0.91)2.427(0.95)2.215(0.92)β14.340(0.94)4.311(0.93)3.745(0.93)β23.654(0.92)3.643(0.92)2.779(0.96)



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*Source: 9929691-2021-06-01.xml*