# Statistical Analysis of Joint Type-I Generalized Hybrid Censoring Data from Burr XII Lifetime Distributions

Authors: Mahmoud Ragab; Aisha Fayomi; Ali Algarni; G. A. Abd-Elmougod; Neveen Sayed-Ahmed; S. M. Abo-Dahab; S. Abdel-Khalek
Journal: Complexity
 (2021)

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## Abstract

The quality of the products coming from different lines of production requires some tests called comparative life tests. For lines having the same facility, the lifetime of the product is distributed by Burr XII, the lifetime distribution, and units are tested under type-I generalized hybrid censoring scheme. The observed censoring data are used under maximum likelihood and the Bayes method to estimate the model parameters. The theoretical results are discussed and assessed through data analysis and Monte Carlo simulation study. Finally, we reported some brief comments obtained from numerical computation.


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## Body

## 1. Introduction

Statistical inference for the life products needs to put some units of product under test to get more information about the life products. Then, we design the life experiments to obtain the required data. Under consideration time and cost, data obtained may be complete or censored. The concept of complete data is used when the failure time of all units under the test is obtained. So far, the concept of censoring data is used when some but not all failure time of units are obtained. Censoring is available in different forms, the oldest ones are called type-I censoring scheme as well as type-II censoring scheme. When we need to run the experiments to prefixed time and the number of units fail is random, the type-I censoring scheme is a suitable scheme. But, when running the experiment to obtain a prefixed number of failure and the total test time is random, type-II censoring scheme is applied. The experimenter in some cases needs to run the experiment under joint cases of type-I and type-II censoring schemes, statistically known with hybrid censoring scheme (HCS).Let the test time is denoted byτ∗ and number of failure units needed to statistical inference is denoted by m, the experiment is removed in hybrid censoring scheme at the only one time of τ∗,Tm. The HCS is combined with type-I and type-II censoring schemes to define type-I and type-II HCS. In type-I HCS, the experiment is removed from the test at the min τ∗,Tm [1, 2], but in type-II HCS, the experiment is removed from the test at the max τ∗,Tm [3]. All of these censoring schemes do not allow terminating units from the test other than the final point; then, it is generalized in progressive censoring scheme which helps us to terminate units at any stage of the experiment, and the key reference of the progressive censoring scheme is given in the study by Balakrishnan and Aggarwala [4]. The two types of censoring schemes are type-I and type-II or the hybrid case; type-I and type-II HCSs have the properties, smaller number of failure may be zero in type-I or total time of the test has a large time in type-II [5]. To overcome this problem [6], the two types of censoring schemes are generalized in generalized hybrid censoring scheme (GHCS) which is described as follows.Type-I GHCS, forn tested units, suppose prior integers s and m that satisfy 1≤s<m≤n and prior ideal test time τ∗∈0,∞. The three cases are considered. If Ts<τ∗, the test is terminated at min τ∗,Tm, and in other cases, if τ∗<Ts<Tm, the test is terminated at Ts, but if tm;n<τ∗, the test is terminated at Tm. The data in type-I GHCS satisfy the minimum number s needing for statistical inference, and the data are summarized as follows:(1)
t¯=t1;n<t1;n<…<ts;n, if τ∗<Ts(2)
t¯=t1;n<t1;n<…<tk;n<…<tr;n, if τ∗>Ts and tm;n>τ∗(3)
t¯=t1;n<t1;n<…<tm;n, if tm;n<τ∗The scheme of the type-I GHCS can be formulated with the schematic diagram described by Figure1.Figure 1
Different types of type-I GHCS.
(a)(b)(c)Type-II GHCS, forn tested units, suppose prior times τ1∗ and τ2∗∈0,∞, such that τ1∗<τ2∗ and integer m satisfies that 1≤m≤n. The three cases are considered. If Tm<τ1∗, the test is terminated at τ1∗, and in second case, if τ1∗<Tm<τ2∗, the test is terminated at Tm. If τ2∗<Tm, the test is terminated at τ2∗. The data in type-II GHCS satisfy the maximum time τ2∗, and the data are summarized as follows:(1)
t¯=t1;n<t1;n<…<tr;n, if Ts<τ1∗ or τ2∗<Tm(2)
t¯=t1;n<t1;n<…<tm;n, if τ1∗<Tm<τ2∗The scheme of the type-II GHCS can be formulated with the schematic diagram described by Figure2.Figure 2
Different types of type-I GHCS.
(a)(b)(c)For manufactured products coming from different lines of production, the problem of determining the relative merits of life products has considerable attention through last view years. Practices, suppose two lines of production are denoted byΓ1 and Γ2 in competing duration and let two independent samples with sizes N1 and N2, respectively. The joint sample of N1 and N2 is put under life testing. This experiment is restricted under consideration of time and cost to terminate after fixed time or number of failure. The data obtained from type of censoring are called joint samples discussed early in [7, 8]. The exact likelihood inference with bootstrap technique under joint sample is presented in [9]. For progressive joint sample, refer studies by Rasouli A and Balakrishnan [10, 11] and recently by B. N. Al-Matrafi and G. A. Abd-Elmougod[12]. Also, for the accelerate model of Rayleigh distribution, refer studies by Faten A. Momenkhan and Abd-Elmougod [13, 14].Type-I GHCS can save time and minimum number needing in statistical inference. So this study aims at development of statistical inference for life products in competing duration under considering type-I GHCS with jointly censoring scheme. Therefore, first, the model formulation under lifetime Burr XII distribution is under jointly type-I GHCS scheme. So far, parameters estimation of Burr XII distributions is carried out when jointly type-I GHCS samples are available. The maximum likelihood and Bayes methods are applied for the parameters estimation. The developed theoretical method assessed through the simulation study as well as illustration is reported with data analysis.The study is planned as follows: the concept and model formulation are reported in Section2. Estimation with maximum likelihood as point and interval estimators is discussed in Section 3. Bayesian approach for point and credible interval estimators with the help of the MCMC method is presented in Section 4. Data analysis is exposed in Section 5. The numerical computation is discussed through a simulation study in Section 6. Finally, some brief comments are reported in Section 7.


## 2. Model

Consider that the product comes from two different lines of productionΓ1 and Γ2 that have the same facility. Suppose, independent two samples of sizes N1 and N2 selected from Γ1 and Γ2 have independent and identical distributed (i.i.d) lifetimes W1,W2,…,WN1 and Z1,Z2,…,ZN2, respectively. The independent samples distributed with populations have fj. probability density functions (PDFs) and Fj. cumulative distribution functions (CDFs) for j=1, 2. The lifetime experiment begins with prior integers and ideal test time given by s,m,τ∗. Through the experiment, the unit failure time, and its type, means from Γ1 and Γ2 are recorded. Then, the experiment is continual until rth failure is observed; if τ∗<Ts, then r=s, but if τ∗>Tm, then r=m, and in other cases, s<r<m. The vector of ordered sample T=T2,δ1,T2,δ1,…,Tr,δr from the sample W1,W2,…,WN1r,Z1,Z2,…,ZN2r with r=N1r+N2r, and the integer r is taken with s and m, or integer that satisfies s<r<m is called joint type-I GHSC. In joint type-I GHSC, δi,i=1,2,…,r, take the value 1 or 0 depending on failure unit from Γ1 or Γ2, respectively, and the two integers m1 and m2 denote the number of units fails from lines Γ1 and Γ2, respectively.The joint likelihood function from observed joint type-I GHSCt = {(t2, δ1), (t2, δ1), …, (tr, δr)} and m1=∑i=1rδi, m2=∑i=1r1−δi is presented by(1)f1,2,…,rt¯=N1!N2!S1DN1−m1S2DN2−m2N1−n1!N2−m2!∏i=1rf1tiδif2ti1−δi,where(2)D=ts,ifτ∗<Tstm,,ifTm<τ∗τ∗ifTs<τ∗<Tm,and survival functionsSj., j=1, 2.Suppose that the lifetimeT has two parameters Burr XII populations with PDFs given by(3)flt=aibitbi−11+tbi−ai+1,t>0,ai>0,bi>0,i=1,2,and CDFs, survival functionsSjt, and failure rate function hit are given by(4)Fit=1−1+tbi−ai,Sit=1+tbi−ai,hit=aibitbi−11+tbi−1,t>0.The function (4) shows that the parameter ai does not affect the shape of failure rate function hit. Burr XII has been applied in areas of quality control, reliability studies, duration, and failure time modeling, see for example [15].


## 3. Estimations under the Maximum Likelihood Method

The joint likelihood function (1) under Burr XII populations (3) and (4) and observed joint type-I GHS sample t=t2,δ1,t2,δ1,tr,δr is formed by(5)La1,b1,a2,b2|t¯∝a1b1m1a2b2m2expb1−1∑i=1rδilogti−a1+1∑i=1rδi×log1+tib1−a1+1N1−m1log1+Db1+b2−1∑i=1r1−δilogti−a2+1∑i=1r1−δilog1+tib1−a2+1N2−m2log1+Db2.The natural logarithms of (5) without a normalized constant reduce to(6)ℓa1,b1,a2,b2|t¯=m1loga1b1+m2loga2b2+b1−1∑i=1rδilogti−a1+1×∑i=1rδilog1+tib1−a1+1N1−m1log1+Db1+b2−1∑i=1r1−δilogti−a2+1∑i=1r1−δilog1+tib1−a2+1N2−m2log1+Db2.


### 3.1. MLEs

Under partial derivative of the log-likelihood function with respect to model parameters and equating to zero, we obtain the likelihood equations as follows:(7)∂ℓa1,b1,a2,b2|t¯∂ai=0,i=1,2,and are reduced to(8)a1b1=m1∑i=1rδilog1+tib1+N1−m1log1+Db1and(9)a2b2=m2∑i=1r1−δilog1+tib2+N2−m2log1+Db2.Also,(10)∂ℓa1,b1,a2,b2|t¯∂bi=0,i=1,2,has presented that(11)m1b1+∑i=1rδilogti−a1+1∑i=1rδitib1logti1+tib1+N1−m1Db1logD1+Db1=0(12)m2b2+∑i=1r1−δilogti−a2+1∑i=1r1−δitib2logti1+tib2+N2−m2Db2logD1+Db2=0.Then, the nonlinear, equations (11) and (12) are reduced after replacing a1 and a2 from (8) and (9) to(13)m1b1+∑i=1rδilogti−m1∑i=1rδilog1+tib1+N1−m1log1+Db1+1∑i=1rδitib1logti1+tib1+N1−m1Db1logD1+Db1=0(14)m2b2+∑i=1r1−δilogti−m2∑i=1r1−δilog1+tib2+N2−m2log1+Db2+1∑i=1r1−δitib2logti1+tib2+N2−m2Db2logD1+Db2=0.The two nonlinear equations presented by (13) and (14) present the likelihood equations of parameters b1 and b2, which are more simple to solve with Newton–Raphson or with fixed point iteration. After obtaining the values b^1 and b^2 from (13) and (14), the values a^1 and a^2 are obtained from (8) and (9). In some cases, if m1=0 or m2=0, the parameter values a1 and b1 or a2 and b2, respectively, are difficult to obtain [16].


### 3.2. Approximate Interval Estimation

The second partial derivatives of log-likelihood function (6) with respect to parameters vector ω=a1,b1,a2.b2 are given by(15)∂2ℓa1,b1,a2,b2|t¯∂ai2=−miai2,i=1,2,∂2ℓa1,b1,a2,b2|t¯∂b12=−m1b12−a1+1∑i=1rδitib1logti21+tib12+N1−m1Db1logD21+Db12,∂2ℓa1,b1,a2,b2|t¯∂b22=−m2b22−a2+1∑i=1r1−δitib2logti21+tib22+N2−m2Db2logD21+Db22,∂2ℓa1,b1,a2,b2|t¯∂a1∂b1=∂2ℓa1,b1,a2,b2|t¯∂b1∂a1=−∑i=1rδitib1logti1+tib1−N1−m1Db1logD1+Db1,∂2ℓa1,b1,a2,b2|t¯∂a1∂a2=∂2ℓa1,b1,a2,b2|t¯∂a2∂a1=∂2ℓa1,b1,a2,b2|t¯∂a1∂b2=∂2ℓa1,b1,a2,b2|t¯∂b2∂a1=0,∂2ℓa1,b1,a2,b2|t¯∂b1∂a2=∂2ℓa1,b1,a2,b2|t¯∂a2∂b1=∂2ℓa1,b1,a2,b2|t¯∂b1∂b2=∂2ℓa1,b1,a2,b2|t¯∂b2∂b1=0,∂2ℓa1,b1,a2,b2|t¯∂a2∂b2=∂2ℓa1,b1,a2,b2|t¯∂b2∂a2=−∑i=1r1−δitib2logti1+tib2−N2−m2Db2logD1+Db2.The Fisher information matrix is defined by minus expectation of second partial derivative of the log-likelihood function. Practice, under a large sample, the Fisher information matrix can be approximate with approximate information matrix. Letη denotes the Fisher information matrix defined by(16)η=−E∂2ℓa1,b1,a2,b2|t¯∂ωi∂ωj,i,j=1,2,3,4,where ω=a1,b1,a2,b2. Then, the approximate information matrix of η denoted by η0 is defined as(17)η0=−∂2ℓa1,b1,a2,b2|t¯∂ωi∂ωj|ω^=a^1,b^1,a^2,b^2,i,j=1,2,3,4.Hence, under asymptotic normality distribution of MLEsa^1,b^1,a^2,b^2 with mean a1,b1,a2,b2 and variance covariance matrix η0−1a^1,b^1,a^2,b^2, 1001−2α% confidence intervals from the model parameter ω=a1,b1,a2,b2 are given by(18)a^1∓zαϵ11,b^1∓zαϵ22,a^2∓zαϵ33,b^2∓zαϵ44,and the vectorϵ11,ϵ22,ϵ33,ϵ44 presents the diagonal of the covariance matrix η0−1a^1,b^1,a^2,b^2, and the value zγ is the percentile of the normal (0,1) with right-tail probability α.


## 3.1. MLEs

Under partial derivative of the log-likelihood function with respect to model parameters and equating to zero, we obtain the likelihood equations as follows:(7)∂ℓa1,b1,a2,b2|t¯∂ai=0,i=1,2,and are reduced to(8)a1b1=m1∑i=1rδilog1+tib1+N1−m1log1+Db1and(9)a2b2=m2∑i=1r1−δilog1+tib2+N2−m2log1+Db2.Also,(10)∂ℓa1,b1,a2,b2|t¯∂bi=0,i=1,2,has presented that(11)m1b1+∑i=1rδilogti−a1+1∑i=1rδitib1logti1+tib1+N1−m1Db1logD1+Db1=0(12)m2b2+∑i=1r1−δilogti−a2+1∑i=1r1−δitib2logti1+tib2+N2−m2Db2logD1+Db2=0.Then, the nonlinear, equations (11) and (12) are reduced after replacing a1 and a2 from (8) and (9) to(13)m1b1+∑i=1rδilogti−m1∑i=1rδilog1+tib1+N1−m1log1+Db1+1∑i=1rδitib1logti1+tib1+N1−m1Db1logD1+Db1=0(14)m2b2+∑i=1r1−δilogti−m2∑i=1r1−δilog1+tib2+N2−m2log1+Db2+1∑i=1r1−δitib2logti1+tib2+N2−m2Db2logD1+Db2=0.The two nonlinear equations presented by (13) and (14) present the likelihood equations of parameters b1 and b2, which are more simple to solve with Newton–Raphson or with fixed point iteration. After obtaining the values b^1 and b^2 from (13) and (14), the values a^1 and a^2 are obtained from (8) and (9). In some cases, if m1=0 or m2=0, the parameter values a1 and b1 or a2 and b2, respectively, are difficult to obtain [16].


## 3.2. Approximate Interval Estimation

The second partial derivatives of log-likelihood function (6) with respect to parameters vector ω=a1,b1,a2.b2 are given by(15)∂2ℓa1,b1,a2,b2|t¯∂ai2=−miai2,i=1,2,∂2ℓa1,b1,a2,b2|t¯∂b12=−m1b12−a1+1∑i=1rδitib1logti21+tib12+N1−m1Db1logD21+Db12,∂2ℓa1,b1,a2,b2|t¯∂b22=−m2b22−a2+1∑i=1r1−δitib2logti21+tib22+N2−m2Db2logD21+Db22,∂2ℓa1,b1,a2,b2|t¯∂a1∂b1=∂2ℓa1,b1,a2,b2|t¯∂b1∂a1=−∑i=1rδitib1logti1+tib1−N1−m1Db1logD1+Db1,∂2ℓa1,b1,a2,b2|t¯∂a1∂a2=∂2ℓa1,b1,a2,b2|t¯∂a2∂a1=∂2ℓa1,b1,a2,b2|t¯∂a1∂b2=∂2ℓa1,b1,a2,b2|t¯∂b2∂a1=0,∂2ℓa1,b1,a2,b2|t¯∂b1∂a2=∂2ℓa1,b1,a2,b2|t¯∂a2∂b1=∂2ℓa1,b1,a2,b2|t¯∂b1∂b2=∂2ℓa1,b1,a2,b2|t¯∂b2∂b1=0,∂2ℓa1,b1,a2,b2|t¯∂a2∂b2=∂2ℓa1,b1,a2,b2|t¯∂b2∂a2=−∑i=1r1−δitib2logti1+tib2−N2−m2Db2logD1+Db2.The Fisher information matrix is defined by minus expectation of second partial derivative of the log-likelihood function. Practice, under a large sample, the Fisher information matrix can be approximate with approximate information matrix. Letη denotes the Fisher information matrix defined by(16)η=−E∂2ℓa1,b1,a2,b2|t¯∂ωi∂ωj,i,j=1,2,3,4,where ω=a1,b1,a2,b2. Then, the approximate information matrix of η denoted by η0 is defined as(17)η0=−∂2ℓa1,b1,a2,b2|t¯∂ωi∂ωj|ω^=a^1,b^1,a^2,b^2,i,j=1,2,3,4.Hence, under asymptotic normality distribution of MLEsa^1,b^1,a^2,b^2 with mean a1,b1,a2,b2 and variance covariance matrix η0−1a^1,b^1,a^2,b^2, 1001−2α% confidence intervals from the model parameter ω=a1,b1,a2,b2 are given by(18)a^1∓zαϵ11,b^1∓zαϵ22,a^2∓zαϵ33,b^2∓zαϵ44,and the vectorϵ11,ϵ22,ϵ33,ϵ44 presents the diagonal of the covariance matrix η0−1a^1,b^1,a^2,b^2, and the value zγ is the percentile of the normal (0,1) with right-tail probability α.


## 4. Bayesian Approach

In this section, we discuss the Bayes estimations of model parameters, point, and credible interval. Bayesian approach needs prior information about the model parameters, which we considered as independent gamma prior, described as follows:(19)ωi∝ωiρi−1expυiωi,i=1,2,3,4,where ω=a1,b1,a2,b2 is the model parameter. The joint prior distribution is given by(20)Ψ∗a1,b1,a2,b2∝∏i=44ωiρi−1expυiωi,i=1,2,3,4.Generally, the posterior distribution from the model parametersΨa1,b1,a2,b2|t¯ is given by(21)Ψa1,b1,a2,b2|t¯=Ψ∗a1,b1,a2,b2×La1,b1,a2,b2|t¯∫0∞∫0∞∫0∞∫0∞Ψ∗a1,b1,a2,b2×La1,b1,a2,b2|t¯da1db1da2db2.And the Bayes estimate under squared error loss function (SEL) of functionΩa1,b1,a2,b2 is given by(22)Ω^B=∫0∞∫0∞∫0∞∫0∞Ωa1,b1,a2,b2Ψa1,b1,a2,b2|t¯da1db1da2db2.The equation (22) has ratio of two integrals, generally cannot be obtained in a closed form. Then, numerical approximation will be used to solve this problem. One way is called numerical integration, and other way used can be called Lindley approximate. The important method which has considerable attention in the last year called the MCMC method is discussed as follows.The proportional form of joint posterior distribution (21) with joint prior distribution (20) and the likelihood function (6) is given by(23)Ψa1,b1,a2,b2|t¯∝a1m1+ρ1−1b1m1+ρ2−1a2m2+ρ3−1b2m2+ρ4−1exp−υ1a1−υ2b1−υ3a2−υ4b2+b1−1∑i=1rδilogti−a1+1∑i=1rδilog1+tib1−a1+1×N1−m1log1+Db1+b2−1∑i=1r1−δilogti−a2+1×∑i=1r1−δilog1+tib1−a2+1N2−m2log1+Db2.The joint posterior distribution (23) reduced with the full-conditional probability distributions PDF’s is given as follows:(24)Ψ1a1|b1,a2,b2,t¯∝exp−a1υ1+∑i=1rδilog1+tib1+N1−m1log1+Db1×a1m1+ρ1−1,Ψ2a2|a1,b1,b2,t¯∝exp−a2υ3+∑i=1r1−δilog1+tib2+N2−m2log1+Db2×a2m2+ρ2−1,Ψ3b1|a1,a2,b2,t¯∝b1m1+ρ2−1exp−υ2b1+b1∑i=1rδilogti−a1+1×∑i=1rδilog1+tib1a1+1N1−m1log1+Db1(25)Ψ4b2|a1,a2,b1,t¯∝b2m2+ρ4−1exp−υ4b2+b2∑i=1r1−δilogti−a2+1×∑i=1r1−δilog1+tib2a2+1N2−m2log1+Db2.The concept of the MCMC method is dependent on the form of the full-conditional distributions and a suitable technique in variety type of MCMC schemes. The posterior distribution is reduced to two gamma distributions (24) as well as two functions (24) and (25), more similar to the normal distribution. Hence, we adopted Gibbs algorithms, and in more general cases, Metropolis Hasting (MH) under Gibbs [17] algorithms.MH under Gibbs algorithms:(1)
The indicatorκ begins with the value 1 and initial parameters vector ω0=a^1,b^1,a^2,b^2(2)
The valuesa1κ and a2κ are generated from gamma distributions (24)(3)
MH algorithms withNb1κ−1,ϵ22 and Nb2κ−1,ϵ44 proposal distributions are used to generate b1κ and b2κ, respectively, as follows(i)
Begin withb^1 and b^2 as an arbitrary starting point b10 and b10 for (24) and (25)(ii)
At timek, sample a candidate point or proposal b1∗, from Nb1κ−1,ϵ22, and b2∗, from Nb2κ−1,ϵ44, the proposal distributions(iii)
Calculate the acceptance probability(26)ρ1b1k−1,b1∗=min1,Ψ3b1∗|a1k−1,a2k−1,b2k−1,t¯fb1k−1|x¯,ρ2b2k−1,b2∗=min1,Ψ4b2∗|a1k−1,a2k−1,b1k−1,t¯fb2k−1|x¯.(iv)
GenerateUi∼U0,1,i=1,2(v)
IfUi≤ρibik−1,bi∗, accept the proposal and set bit=bi∗. Otherwise, reject the proposal and set bik=bik−1.(4)
Report the simulate parameters vectorωκ=a1κ,b1κ,a2κ,b2κ(5)
Changeκ to κ+1(6)
Repeat steps from 2 to5, M times.Bayes estimation with the MCMC method requires some measurements reported about the generation method and determine the number needed to reach the stationary distribution (burn-in) denoted byM∗. Then, the Bayes estimators are given by(27)Ω^B=EΨΩ|t¯=1M−M∗∑i=M∗+1MΩi.Also, the posterior variance of functionΩ is given by(28)V^Ω|t¯=1M−M∗∑i=M∗+1MΩi−Ω^B2.The credible interval is obtained with ordering the simulated vectorΩ; then, 1001−2α% credible interval of function Ω is given by(29)ΩαM−m∗,Ω1−αM−M∗.


## 5. Illustrative Example

In this section, we summarized a simulated dataset from two Burr XII populations to check the theoretical results discussed through the study. The real parameters value is selected to satisfy prior information, so that withρ1,ρ2,ρ3,ρ4=2,3,4,2 and υ1,υ2,υ3,υ4=2,2.5,2,2, the real parameter values are selected to satisfy ωi≃EΨ∗ωi=ρi/υi. Then, the parameters values are selected to be ω=a1,b1,a2,b2=1.82,1.7,1.2,1.92. Under Burr XII distribution, parameter values (1.82, 1.7) generate a sample of size N1=30 to present data from the line Γ1 that are given by W = { 0.0604831, 0.167157, 0.268757, 0.284369, 0.336897, 0.389696, 0.402109, 0.434761, 0.539457, 0.548363, 0.569382, 0.584881, 0.604739, 0.607685, 0.640021, 0.719833, 0.747014, 0.752982, 0.760519, 0.789178, 0.848233, 0.885767, 0.892944, 0.912513, 0.931659, 0.939561, 1.21948, 1.49421, 1.50994, 1.59295}. Also, Burr XII distribution with parameters (1.2, 1.92) generates a sample of size N2=30 to present data from the line Γ2 that are given by Z = {0.0459775, 0.156614, 0.293896, 0.337772, 0.397422, 0.446852, 0.466979, 0.469356, 0.470276, 0.475754, 0.499894, 0.596894, 0.761898, 0.766247, 0.824323, 0.922044, 0.940813, 0.968182, 0.975155, 0.975229, 1.0076, 1.18989, 1.21431, 1.62018, 1.74474, 1.82949, 1.98172, 2.59949, 2.76045, 5.17082}. Therefore, for given prior integers s,m=25,35 and the prior time τ∗=0.5, the joint type-I GHC data are summarized in Table 1. In the last sections, we discussed two different methods of estimation, MLEs and Byes estimations. The point results of MLE and Bayes estimates as well as asymptotic confidence interval and credible interval are summarized in Table 2. The chen in MCMC methods is reported for 11000 iterations that contain the first 1000 samples as burn-in. Usually, it is not hard to construct a Markov chain with the desired properties. To determine how many steps are needed to converge to the stationary, more difficult problem is distribution within an acceptable error. Then, we can test if stationary distribution is reached quickly starting from an arbitrary position. The plot for the simulation number of the model parameters and the corresponding histogram shown in Figures 3–6 can be used to describe the convergence results in MCMC methods.Table 1
The joint type-I GHS data withs,m=25,35 and τ=0.5, d = 25.
0.04597750.06048310.1566140.1671570.2687570.2843690.29389601011100.3368970.3377720.3896960.3974220.4021090.4347610.44685210101100.4669790.4693560.4702760.4757540.4998940.5394570.54836300000110.5693820.5848810.5968940.6047390.6076850.6400210.71983311011110.7470140.7529820.7605190.7618980.7662470.7891780.8243231110010Table 2
The point and 95% confidence intervals (ACIs and CIs) of MLEs Bayes estimates.
Pa.s(.)ML(.)BMCMC95% ACIsLength95% CILengtha1=1.822.447891.86465(1.1898, 3.7060)2.5163(1.1225, 2.8007)1.6783b1=1.72.446432.14852(1.5600, 3.3329)1.7729(1.4706, 2.9120)1.4413a2=1.22.024581.39188(0.8110, 3.2382)2.4272(0.8109, 2.1501)1.3392b2=1.921.564131.82341(0.8581, 2.2702)1.4121(1.1559, 2.5984)1.4425Figure 3
Simulation number and the corresponding histogram ofa1 generated by the MCMC method.
(a)(b)Figure 4
Simulation number and the corresponding histogram ofb1 generated by the MCMC method.
(a)(b)Figure 5
Simulation number and the corresponding histogram ofa2 generated by the MCMC method.
(a)(b)Figure 6
Simulation number and the corresponding histogram ofa2 generated by the MCMC method.
(a)(b)


## 6. Simulation Studies

The quality of estimators dependent on some tolls or measures that are computed for a suitable numbers of generated samples from the populations with given parameter values is known by a simulation study. Then, we assess the theoretical estimation results of MLEs and Bayes estimators under discussing and computing average (AV) and MSEs for point estimate and coverage probability (CP) and average of interval length (AL) to the interval estimation. The simulation study is reported for different sample sizes (N1+N2) and different effected sample sizes (m). Also, we consider different cases of min number s and different ideal test time (τ∗). Also, we study the effect of parameters change with considering two sets of populations parameters, say ω=a1, b1,a2, b2∈{(1.0, 2.0, 2.0, 3.0), (0.4, 1.2, 1.5, 2.0)}. The prior parameters are selected to satisfy EΨ∗ωi≃ρi/υi. Hence, in our simulation study, we proposed different two cases from prior information; one of them is expressed to noninformative prior (prior0), in which the posterior distribution is proportional with likelihood function. The second case is expressed to informative prior (prior1 and prior2). The prior1 is (ρ1, ρ2, ρ3, ρ4) = (1, 3, 4, 5) and (υ1, υ2, υ3, υ4) = (1, 2,1.5, 2). The prior2 is (ρ1, ρ2, ρ3, ρ4) = (1, 2.5, 3, 4) and (υ1, υ2, υ3, υ4) = (2.0, 2.0, 2.0, 1.5). For Bayesian approach, without loss of the generality for any loss function, all computations are reported under squared error loss function. The MCMC method is performed under 11000 chen with 1000 burn-in, and the results are reported in Tables 3–6.Table 3
The AVs and MSEs of Burr XII populations withω=(1.0, 2.0, 2.0, 3.0).
τ∗N1,N2k,mPa.MLEsBayesprior0Bayesprior1AVsMSEsAVsMSEsAVsMSEs0.5(20, 20)(15, 20)a11.2320.3241.2410.3271.2100.210b12.2130.4102.2300.4072.2000.332a22.1880.4332.2010.3992.2140.341b23.2410.6213.2450.6183.2140.540(15, 30)a11.2010.3001.2220.3091.2020.198b12.2000.3892.2150.3982.1870.301a22.1740.4142.1980.4012.2010.317b23.2150.5983.2320.6003.2200.525(25, 30)a11.1870.2651.1820.2631.1790.154b12.1990.3652.2000.3632.1550.277a22.1420.4012.1510.3952.1880.298b23.2020.5753.2250.5803.2020.478(30, 30)(25, 35)a11.1600.2301.1770.2251.1610.115b12.1880.3322.1900.3372.1410.238a22.1300.3842.1350.3772.1750.265b23.1890.5323.1950.5273.1750.422(25, 50)a11.1440.2181.1520.2211.1490.100b12.1460.3112.1550.3142.1220.219a22.1220.3732.1300.3692.1650.240b23.1700.5113.1880.5143.1640.4011.2(20, 20)(15, 20)a11.1800.2801.1900.2501.2000.170b12.1900.3702.2100.3612.1800.282a22.1650.3422.1810.3442.120.281b23.2250.5703.2250.5713.2000.487(15, 30)a11.1820.2511.2090.2621.1870.152b12.1830.3492.1860.3502.1770.255a22.1560.3642.1620.3502.1870.266b23.2000.5513.2190.5543.2010.470(25, 30)a11.1710.2141.1690.2221.1560.109b12.1790.3182.1700.3162.1420.231a22.1240.3572.1440.3422.1600.240b23.1790.5283.2110.5423.1800.436(30, 30)(25, 35)a11.1460.2311.1330.1871.1460.086b12.1650.2902.1690.3002.1240.187a22.1110.3412.1190.3302.1690.221b23.1800.4803.1800.4873.1610.387(25, 50)a11.1270.2691.1370.1701.1410.060b12.1400.2652.1410.2702.1010.165a22.1020.3232.1190.3272.1480.200b23.1510.4593.1670.4693.1480.366Table 4
The ALs and PCs of 95% CI of Burr XII populations withω=(1.0, 2.0, 2.0, 3.0).
τ∗N1,N2k,m,Pa.MLEsBayesprior0Bayesprior1ALsPCsALsPCsALsPCs0.5(20, 20)(15, 20)a12.513(90)2.490(91)2.122(91)b14.124(89)4.115(90)3.562(90)a24.002(90)4.011(91)3.562(92)b24.865(90)4.800(91)4.142(91)(15, 30)a12.475(90)2.462(91)2.086(92)b14.100(91)4.085(92)3.501(93)a23.992(91)4.001(91)3.539(93)b24.851(90)4.730(93)4.103(94)(25, 30)a12.438(90)2.433(92)2.010(92)b14.050(93)4.040(92)3.470(92)a23.943(91)3.925(90)3.500(93)b24 .801(92)4.750(93)4.009(95)(30, 30)(25, 35)a12.401(90)2.403(92)1.777(94)b13.950(93)3.941(93)3.401(93)a23.900(90)3.890(92)3.420(93)b24.751(93)4.735(93)3.889(93)(25, 50)a12.370(93)2.362(92)1.715(95)b13.901(92)3.902(95)3.360(93)a23.850(92)3.856(92)3.381(94)b24.714(95)4.701(94)3.811(94)1.2(20, 20)(15, 20)a12.480(91)2.475(91)2.085(92)b14.100(90)4.190(90)3.541(90)a23.980(91)3.986(91)3.533(92)b24.838(90)4.801(91)4.112(91)(15, 30)a12.443(91)2.441(93)2.052(95)b14.072(93)4.066(92)3.471(93)a23.961(91)3.955(93)3.512(93)b23.721(91)3.718(93)3.003(94)(25, 30)a12.411(92)2.409(92)1.980(95)b14.012(93)4.007(93)3.441(92)a23.922(92)3.905(93)3.471(95)b24.790(95)4.727(93)4.000(95)(30, 30)(25, 35)a12.370(92)2.381(93)1.744(94)b13.935(93)3.922(93)3.380(93)a23.871(94)3.865(93)3.400(93)b24.733(93)4.725(93)3.842(95)(25, 50)a12.352(94)2.344(92)1.652(97)b13.872(94)3.881(95)3.340(94)a23.823(92)3.826(93)3.354(96)b24.692(95)4.680(94)3.800(94)Table 5
The AVs and MSEs of Burr XII populations withω=(0.4, 1.2, 1.5, 2.0).
τ∗N1,N2k,m,Pa.MLEsBayesprior0Bayesprior1AVsMSEsAVsMSEsAVsMSEs0.4(20, 20)(15, 20)a10.5890.1210.5850.1220.5640.091b11.4440.3711.4400.3651.3990.277a21.5450.3751.5220.3701.5390.309b22.3310.4322.3270.4362.2310.332(15, 30)a10.5620.0880.5550.0900.5340.068b11.4130.3151.4170.3241.3290.227a21.5410.3401.5380.3381.5300.282b22.3100.3992.2990.3952.2310.281(25, 30)a10.5220.0710.5310.0680.5310.049b11.3950.3111.4020.3071.2990.211a21.5350.3181.5310.3251.5370.269b22.3010.3622.2990.3682.2280.268(30, 30)(25, 35)a10.4890.0610.4990.0670.4880.035b11.3810.2771.3880.2691.2900.210a21.5250.3031.5210.3011.5450.255b22.2800.3552.2750.3380.2150.261(25, 50)a10.4710.0440.4600.0410.4660.022b11.3580.2641.3660.2611.2710.181a21.5270.2991.5360.3001.5360.256b22.2720.3442.2660.3512.2250.2511.1(20, 20)(15, 20)a10.5620.0850.5550.0820.5320.061b11.4210.3311.4120.3281.3850.238a21.5240.3451.5110.3411.5180.284b22.3120.4102.3150.4180.2100.299(15, 30)a10.5410.0760.5380.0720.5200.049b11.3990.3021.4020.3071.3180.211a21.5220.3211.5210.3181.5000.265b22.2910.3842.2880.3780.2130.272(25, 30)a10.5010.0570.5050.0580.5180.028b11.3780.2871.4000.2901.2870.199a21.5170.3001.5190.2981.5130.251b22.2880.3452.2780.3390.2110.247(30, 30)(25, 35)a10.4790.0420.4800.0380.4770.017b11.3660.2601.3700.2551.2770.175a21.5120.2981.5110.2851.5300.241b22.2670.3322.2400.3380.2100.247(25, 50)a10.4550.0370.4450.0290.4540.012b11.3450.2471.3360.2391.2590.164a21.5100.2841.5300.2771.5200.241b22.2670.3322.2400.3380.2100.239Table 6
The ALs and PCs of 95% CI of Burr XII populations withω=(0.4, 1.2, 1.5, 2.0).
τ∗N1,N2k,m,Pa.MLEsBayesprior0Bayesprior1ALsPCsALsPCsALsPCs0.4(20, 20)(15, 20)a11.752(89)1.744(89)1.610(90)b13.245(90)3.260(90)2.452(91)a23.745(88)3.749(89)3.541(90)b24.452(89)4.445(90)4.223(90)(15, 30)a11.700(90)1.702(89)1.564(91)b13.201(90)3.212(92)2.403(92)a23.700(90)3.703(90)3.500(92)b24.411(91)4.402(90)4.185(92)(25, 30)a11.640(92)1.652(90)1.511(93)b13.153(92)3.280(92)2.379(92)a23.640(92)3.632(92)3.470(94)b24.382(91)4.379(93)4.162(93)(30, 30)(25, 35)a11.612(95)1.624(94)1.480(93)b13.122(92)3.145(92)2.344(96)a23.611(92)3.608(96)3.444(94)b24.324(94)4.333(93)4.141(94)(25, 50)a11.514(95)1.502(94)1.432(93)b13.100(92)3.104(92)2.312(96)a23.581(92)3.575(96)3.402(94)b24.284(94)4.279(93)4.113(94)1.1(20, 20)(15, 20)a11.532(89)1.539(90)1.425(90)b13.001(90)0.297(90)2.254(91)a23.521(90)3.535(91)3.324(92)b24.213(89)4.225(90)4.001(91)(15, 30)a11.512(90)1.518(91)1.411(90)b12.985(91)2.978(90)2.233(91)a23.503(90)3.501(91)3.303(92)b24.188(91)4.177(91)3.987(92)(25, 30)a11.480(92)1.475(93)1.324(92)b12.966(91)2.950(91)2.270(94)a23.480(92)3.475(93)3.275(93)b24.125(93)4.135(92)3.944(93)(30, 30)(25, 35)a11.435(90)1.422(93)1.300(96)b12.945(93)2.231(92)2.217(94)a23.444(93)3.442(93)3.241(95)b24.109(93)4.100(94)3.901(92)(25, 50)a11.401(92)1.398(94)1.241(94)b12.911(93)2.215(93)2.200(94)a23.414(95)3.411(93)3.223(93)b24.045(93)4.055(91)3.840(93)


## 7. Concluding Remarks

In the industrial field, the existing different lines of production have the same products under the same facility. The problem of measuring the relative merits of product in the competing duration has considerable attention in past view years. This problem has been discussed in this study for products distributed with Burr XII lifetime distribution. This problem presented in parameters estimation forms with ML and Bayesian estimations under joint type-I GHCS. Then, the developed method is assessed through the Monte Carlo simulation study. The results obtained from these studies show the following comments.(1)
All results obtained in Tables3–6 show that the developed method works well in all cases under joint type-I GHCS for Burr XII lifetime products(2)
The results under MLE and noninformative Bayes estimation are closed to itself(3)
The Bayes method performs better than the ML method under informative prior(4)
The MSEs and interval length are reduced under increases in effective sample sizes,m(5)
The results perform better for the large value of test timeτ∗(6)
Under parameters change, the results of the simulation study show the validly of the results of all parameters chosen



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