# Statistical Inferences of Burr XII Lifetime Models under Joint Type-1 Competing Risks Samples

Authors: Tahani A. Abushal; A. A. Soliman; G. A. Abd-Elmougod
Journal: Journal of Mathematics
 (2021)

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## Abstract

The problem of statistical inference under joint censoring samples has received considerable attention in the past few years. In this paper, we adopted this problem when units under the test fail with different causes of failure which is known by the competing risks model. The model is formulated under consideration that only two independent causes of failure and the unit are collected from two lines of production and its life distributed with Burr XII lifetime distribution. So, under Type-I joint competing risks samples, we obtained the maximum likelihood (ML) and Bayes estimators. Interval estimation is discussed through asymptotic confidence interval, bootstrap confidence intervals, and Bayes credible interval. The numerical computations which described the quality of theoretical results are discussed in the forms of real data analyzed and Monte Carlo simulation study. Finally, numerical results are discussed and listed through some points as a brief comment.


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## Body

## 1. Introduction

The failure times which are obtained from life testing experiments are exposed in complete or censored data. Therefore, the word complete data is used when the failure time of all units under the test is observed but, under some restrictions of time and cost, the failure time of some not all units is observed. Then, we used the word censoring data when the available lifetime data are taken from some units under the test. Censoring scheme can be done under different forms, and the commonly ones are known by Type-I and Type-II censoring schemes (CSs). In Type-I CS, the test has a prefixed time and random number of failure units. However, in Type-II CS, the test time is random and has prefixed number of failure units. Each of Type-I CS and Type-II CS does not allow to remove unit from the test other than the final point. The availability of removed units from the test at any stage is known by progressive censoring scheme (see Balakrishnan and Aggarwala [1]). Under consideration that units of product are taken from different lines of production under the same facility, the joint censoring scheme appeared. Censoring schemes under joint sample are called joint censoring scheme (JCS). Therefore, we combine the joint censoring scheme with Type-I and Type-II censoring schemes to obtain the Type-I and Type-II joint censoring schemes (Type-I and Type-II JCSs).The product produced from the different lines of production under the same facilities needs some tests to measure the relative merits in a competing duration. In practice, JCSs are applied on random selection taken from lines of production. Different authors exposed to this problem, for early discussion, such as Rao et al. [2] developed the rank order theory under two-sample censoring scheme, Basu [3] presented and discussed the statistics of rank sets from two-sample scheme called Savage statistic, Johnson and Mehrotra [4] used two-sample problem to preset the locally most powerful rank tests under censored data, Bhattacharyya and Johnson [5] applied two-sample censored situation for asymptotic sufficiency and asymptotically most powerful tests, Mehrotra and Bhattacharyya [6] measured the equality of two exponential distributions testing under Type-II censoring, and Mehrotra and Bhattacharyya [7] discussed under jointly Type-II censored samples the confidence intervals from two exponential distributions. Also, Balakrishnan and Rasouli [8] presented exact likelihood inferences under jointly censoring schemes, Rasouli and Balakrishnan [9] discussed the exact likelihood inference under joint progressive Type-II censoring for two exponential populations, and Shafay et al. [10] discussed the Bayes inference under joint Type-II censored sample for two exponential populations. And, this problem is handled recently by Al-Matrafi and Abd-Elmougod [11], Momenkhan and Abd-Elmougod [12], Mondal and Kundu [13], and Mondal andKundu [14]. The problem of statistical inference under jointly censoring schemes with the competing risks model is recently discussed by Almarashi et al. [15].Under Type-I JCS, a sample of sizeN is randomly selected from two lines of production η1 and η2 to satisfy that S1 is selected from the first line η1 and S2 is selected from the second line η2, and the ideal test time τ is given. The sample of size S1 taken from the line η1 has T1,T2,…,TS1 lifetimes distributed with PDF and CDF given, respectively, by f1⋅ and F1⋅. Also, S2 from the line η2 has T1,T2,…,TS2 lifetimes with PDF and CDF given, respectively, by f2⋅ and F2⋅. Under given the test time τ, the ordered life times X1,X2,…,XJ, 1≤J≤N, obtained from the joint sample T1,T2,…,TJ1,T1,T2,…,TJ2, J=J1+J2, present the Type-I JCS. Therefore, under Type-I JCS, the failure time and the corresponding type of failure (mean from the line η1 or η2) are recorded. Hence, the Type-I JCS is given by(1)X=X1,υ1,X2,υ2,…,XJ,υJ,where υi=1 or 0 dependent on the failure from the line η1 or the line η2, respectively. Suppose that the integer numbers denoted the number of failure from the line η1 given by n1=∑i=1Jυi and number of failure from the line η2 given by n2=∑i=1J1−υi. Hence, the joint likelihood function under X, Type-I JCS, is formulated by(2)L1,2,…,JX|ω=S1!S2!S1−n1!S2−n2!∏i=1Jf1xiυif2xi1−υiR1S1−n1xJR2S2−n2xJ,where Ri⋅, i=1,2, mean the reliability functions and ω presents the parameters vector.In a real-life testing, commonly the failure times of units/individuals may be reported under different causes of failure which is known by the competing risks model. In this problem, our aim is measuring the risk of one cause of failure with respect to other causes. Early, this problem was discussed under exponential populations by Cox [16] and some properties of the competing risks model by Crowder [17], Balakrishnan and Han [18], Modhesh and Abd-Elmougod [19], and Bakoban and Abd-Elmougod [20]. Recently, the properties of the competing risks model under the accelerated life test model were discussed by Ganguly and Kundu [21], Hanaa and Neveen [22], and Algarn et al. [23]. The competing risk problem under Type-I censoring scheme can be described as follows.Suppose thatN unit is put under life testing experiment and the ideal test τ is given under consideration that only two independent causes of failure exist. The failure time and the corresponding cause of failure are recorded, say X=X1,δ1,X2,δ2,…,XJ,δJ and 1≤J≤N. The joint likelihood function under competing risks Type-I,X, is formulated by(3)LX|ω=n!n−J!∏i=1Jh1xiμδi=1h2xiμδi=2R1xiR2xiR1xJR2xJN−J,where(4)μδi=l=1,δi=l,0,δi≠l,l=1,2,(5)0<x1<x2<⋯<xJ<∞.Early, the Burr system is introduced as a system that includes twelve types of cumulative distribution functions (see Burr [24]). Also, the Burr system present a variety of density shapes that are applied in different branches of sciences such as chemical engineering, medical and reliability studies, business, and quality control. The Burr XII distribution which is member of this system has different application in life testing models. The random variable X is called Burr XII random variable if it has cumulative distribution function (CDF) given by(6)Fx=1−1+xβ−α,x>0.Burr XII distribution has unimodal or decreasing failure rate function. Also, the shape of failure rate function is not affected by shape parametersα and has unimodal curve when β>1. Also, it has decreasing failure rate function when β≤1. Therefore, the shape parameter β is more effective in distribution. Different authors discussed Burr XII such as Rodriguez [25], Lee et al. [26], and recently Hassan and Nada [27].The product coming from different lines of production is tested under the type of testing known by comparative life tests. When population units or individuals fail under different causes of failure, we have joint competing risks' data as an important source of data. Our aims in this paper are building the statistical inferences of Burr XII life populations based on this competing risk Type-I JCS. Then, we give a complete description for the model formulation considering only two independent causes of failure and the unit life distributed with Burr XII lifetime distribution. The collected data observed under this model are used to estimate the model parameters with maximum likelihood estimation for point and corresponding confidence interval. Also, two confidence intervals with bootstrap-p and bootstrap-t are formulated. The Bayes approach is used to construct the point and credible interval estimations. Different tools are used to measure the quality performance of these estimators. The point estimations were measured under mean squared errors (MSEs). And, the interval estimations were measured under interval length (IL) and probability coverage (PC) through the Monte Carlo simulation study. Also, we analyze the real data set to illustrate our purpose.The paper is planned as follows. Section2 discusses general assumptions and modeling. Estimation with MLE, point, and asymptotic confidence intervals is presented in Section 3. Bootstrap confidence intervals are discussed in Section 4. Bayes estimation is presented in Section 5. The real example is used and analyzed in Section 6. Assessment and comparing the numerical results with simulation study are presented in Section 7. The brief comments are summarized in Section 8.


## 2. Model Formulation

Let a sample of sizeN=S1+S2 be selected from two lines η1 and η2 (S1 from η1 and S2 from η2) for a life testing experiment, and the ideal test time τ is proposed. When the experiment is running, the failure time X and the corresponding type υ as well as cause of failure δ are reported. The experiment is continual until τ is observed; then, we can say Xi,υi,δi, i=1,2,…,J, are observed. Therefore, the random set X=X1,υ1,δ1,X2,υ2,δ2,…,XJ,υJ,δJ, and 1≤J≤N is called Type-I joint competing risks sample (Type-I JCRS). Therefore, under Type-I JCRS, we have the following assumption:(1)
The numbern1=∑i=1Jυi present number of failure from the line η1.(2)
The numbern2=∑i=1J1−υi present number of failure from the line η2.(3)
The numberm1j=∑i=1Jυi∗μδi=j present number of failure from the line η1 and cause j.(4)
The numberm2j=∑i=1J1−υi∗μδi=j present number of failure from the line η2 and cause j. Hence, the joint likelihood function of Type-I JCRS X=X1,υ1,δ1,X2,υ2,δ2,…,XJ,υJ,δJ is formulated by(7)L1,2,…,JX|ω∝∏i=1Jh11xiμδi=1h12tiμδi=2R11xiR12xiυi×h21xiμδi=1h22tiμδi=2R21xiR22xi1−υi×R11xJR12xJS1−n1R21xJR22xJS2−n2,whereμδi=l is given by (4).(5)
Ifk defines the unit type, then the observed failure time xi=minxik1,xik2, i=1,2,…,J.(6)
The CDF of random variableXikj of Burr XII lifetime distribution is given by(8)Fkjx=1−1+xβk−αkj,x>0,βk,αkj>0,k,j=1,2.(7)
The minimum value has distribution given byFk1⋅+Fk2⋅−Fk1⋅∗Fk2⋅. Therefore, the latent failure time is distributed with Burr XII distributions with shape parameters βk and αk1+αk2.(8)
The discrete random variablesm1j and m2j have the binomial distributions given by(9)m1j⟶binomialnj,αk1αk1+αk2,m2j⟶binomialnj,αk2αk1+αk2.


## 3. Maximum Likelihood Estimation

The model parameters in this section are discussed under given Type-I JCRS from Burr XII distribution. The joint likelihood function (7) is reduced to(10)Lω|X∝∏i=1Jxiβ1−11+xiβ1−α11+α12+1υixiβ2−11+xiβ2−α21+α22+11−υi×1+xJβ1−S1−n1α11+α121+xJβ2−S2−n2α21+α22×β1n1β2n2α11m11α12m12α21m21α22m22xJβ1−1xJβ2−1,where ω=β1,β2,α11,α12,α21,α22 and X be Type-I JCRS. Function (10) after taking the natural logarithm is reduced to(11)ℓω|X=n1logβ1+n2logβ2+m11logα11+m12logα12+m21logα21+m22logα22+β1−1∑i=1Jυilogxi−α11+α12+1∑i=1Jυilog1+xiβ1+β2−1∑i=1J1−υilogxi−α21+α22+1∑i=1J1−υilog1+xiβ2−S1−n1α11+α12log1+xJβ1−S2−n2α21+α22log1+xJβ2.


### 3.1. Point Estimation

From the log-likelihood function, we obtain the likelihood equations by taking the first partially derivatives respective to the model parameters as follows:(12)∂ℓω|X∂α1j=m1jα1j−∑i=1Jυilog1+xiβ1−S1−n1log1+xJβ1=0,∂ℓω|X∂α2j=m2jα2j−∑i=1J1−υilog1+xiβ2−S2−n2log1+xJβ2=0,which reduced to(13)α^1jβ1=m1j∑i=1Jυilog1+xiβ1+S1−n1log1+xJβ1,(14)α^2jβ2=m2j∑i=1J1−υilog1+xiβ2+S2−n2log1+xJβ2.And the derivatives with respect toβk are reduced to the likelihood equations as follows:(15)∂ℓω|X∂βk=0,k=1,2,which reduced to(16)n1β1+∑i=1Jυilogxi−α11+α12+1∑i=1Jυixiβ1logxi1+xiβ1−S1−n1α11+α12xJβ1logxJ1+xJβ1=0,(17)n2β2+∑i=1J1−υilogxi−α21+α22+1∑i=1Jυixiβ2logxi1+xiβ2−S2−n2α21+α22xJβ2logxJ1+xJβ2=0.Equations (13) to (17) have shown that the problem of obtaining the ML estimate of model parameters needs to solve two nonlinear equations (16) and (17) to obtain β^k, k=1,2. Different iteration methods can be applied such as Newton–Raphson or fixed point iteration with initial value can be obtained from the profile log-likelihood (11) after replacing the parameters αkj of equations (13) and (14) as follows:(18)fβ1,β2|X=n1logβ1+n2logβ2+m11logα^11β1+m12logα^12β1+m21logα^21β2+m22logα^22β2+β1−1∑i=1Jυilogxi−α^11β1+α^12β1+1∑i=1Jυilog1+xiβ1+β2−1×∑i=1J1−υilogxi−α^21β2+α^22β2+1∑i=1J1−υi×log1+xiβ2−S1−n1α^11β1+α^12β1log1+xJβ1−S2−n2α^21β2+α^22β2log1+xJβ2.Also, the ML estimate of parametersα^kj is obtained from (13) and (14) after replacing βk by β^k.Remark 1.
The equations from (13) to (17) showed that the conditional estimators of the model parameters depend on the discrete random variable mkj. Hence, the estimate α^1j and α^2j does not exist for m1j=0 or J and m2j=0 or J, respectively. And, the problem of exact distributions for estimators α^1j and α^2j is defined as mixture of discrete and continuous distributions, hence as given in Kundu and Joarder [28] is difficult to obtain.


### 3.2. Interval Estimation

The asymptotic confidence intervals of model parameters depend on the second partial derivative of the log-likelihood function (11) and hence information matrix (see Salah [29]). And, the Fisher information matrix of the model parameters is defined as the minus expectation of the second partial derivatives which is presented as follows:(19)∂2ℓω|X∂β12=−n1β12−α11+α12+1∑i=1Jυixiβ1logxi21+xiβ12−S1−n1α11+α12xJβ1logxJ21+xJβ12,∂2ℓω|X∂β22=−n2β22−α21+α22+1∑i=1J1−υixiβ2logxi21+xiβ22−S1−n1α21+α22xJβ2logxJ21+xJβ22,∂2ℓω|X∂αkj2=−mkjαkj2|k,j=1,2,∂2ℓω|X∂αkj∂αil=0,For eachkj≠il,∂2ℓω|X∂β1∂α1j=∂2ℓω|X∂α1j∂β1=−∑i=1Jυixiβ1logxi1+xiβ1−S1−n1xJβ1logxJ1+xJβ1,j=1,2,∂2ℓω|X∂β2∂α2j=∂2ℓω|X∂α2j∂β2=−∑i=1J1−υixiβ2logxi1+xiβ2−S2−n2xJβ2logxJ1+xJβ2,j=1,2,∂2ℓω|X∂β1∂α2j=∂2ℓω|X∂β2∂α1j=∂2ℓω|X∂α2j∂β1=∂2ℓω|X∂α1j∂β2=∂2ℓω|X∂β1∂β2=∂2ℓω|X∂β2∂β1=0.Suppose that the fisher information matrix is defined byΨβ1,β2,α11,α12,α21,α22, where(20)Ψβ1,β2,α11,α12,α21,α22=−E∂2ℓω|X∂ωi∂ωl,i,l=1,2,…,6,where ω=β1,β2,α11,α12,α21,α22 be the model parameters. Equation (19) has shown that the expectations of the second derivative of the log likelihood function are more serious. Therefore, we applied the approximate information matrix Ψ^0β^1,β^2,α^11,α^12,α^21,α^22 defined by(21)Ψ^0β^1,β^2,α^11,α^12,α^21,α^22=∂2ℓω|X∂ωi∂ωlβ^1,β^2,α^11,α^12,α^21,α^22,i,l=1,2,…,6.Therefore,Ψ^0−1β^1,β^2,α^11,α^12,α^21,α^22 exists with nonzero values of the elements of diagonal. Under normal properties of β^1,β^2,α^11,α^12,α^21,α^22, the approximate 1−2θ% confidence intervals of the parameters β1,β2,α11,α12,α21, and α22 are given by(22)β^1∓zθϵ11,β^2∓zθϵ22,α^11∓zθϵ33,α^12∓zθϵ44,α^21∓zθϵ55,α^22∓zθϵ66,where ϵil is the element of diagonal of the invariance approximate information matrix Ψ^0−1β^1,β^2,α^11,α^12,α^21,α^22 with significant level θ.


## 3.1. Point Estimation

From the log-likelihood function, we obtain the likelihood equations by taking the first partially derivatives respective to the model parameters as follows:(12)∂ℓω|X∂α1j=m1jα1j−∑i=1Jυilog1+xiβ1−S1−n1log1+xJβ1=0,∂ℓω|X∂α2j=m2jα2j−∑i=1J1−υilog1+xiβ2−S2−n2log1+xJβ2=0,which reduced to(13)α^1jβ1=m1j∑i=1Jυilog1+xiβ1+S1−n1log1+xJβ1,(14)α^2jβ2=m2j∑i=1J1−υilog1+xiβ2+S2−n2log1+xJβ2.And the derivatives with respect toβk are reduced to the likelihood equations as follows:(15)∂ℓω|X∂βk=0,k=1,2,which reduced to(16)n1β1+∑i=1Jυilogxi−α11+α12+1∑i=1Jυixiβ1logxi1+xiβ1−S1−n1α11+α12xJβ1logxJ1+xJβ1=0,(17)n2β2+∑i=1J1−υilogxi−α21+α22+1∑i=1Jυixiβ2logxi1+xiβ2−S2−n2α21+α22xJβ2logxJ1+xJβ2=0.Equations (13) to (17) have shown that the problem of obtaining the ML estimate of model parameters needs to solve two nonlinear equations (16) and (17) to obtain β^k, k=1,2. Different iteration methods can be applied such as Newton–Raphson or fixed point iteration with initial value can be obtained from the profile log-likelihood (11) after replacing the parameters αkj of equations (13) and (14) as follows:(18)fβ1,β2|X=n1logβ1+n2logβ2+m11logα^11β1+m12logα^12β1+m21logα^21β2+m22logα^22β2+β1−1∑i=1Jυilogxi−α^11β1+α^12β1+1∑i=1Jυilog1+xiβ1+β2−1×∑i=1J1−υilogxi−α^21β2+α^22β2+1∑i=1J1−υi×log1+xiβ2−S1−n1α^11β1+α^12β1log1+xJβ1−S2−n2α^21β2+α^22β2log1+xJβ2.Also, the ML estimate of parametersα^kj is obtained from (13) and (14) after replacing βk by β^k.Remark 1.
The equations from (13) to (17) showed that the conditional estimators of the model parameters depend on the discrete random variable mkj. Hence, the estimate α^1j and α^2j does not exist for m1j=0 or J and m2j=0 or J, respectively. And, the problem of exact distributions for estimators α^1j and α^2j is defined as mixture of discrete and continuous distributions, hence as given in Kundu and Joarder [28] is difficult to obtain.


## 3.2. Interval Estimation

The asymptotic confidence intervals of model parameters depend on the second partial derivative of the log-likelihood function (11) and hence information matrix (see Salah [29]). And, the Fisher information matrix of the model parameters is defined as the minus expectation of the second partial derivatives which is presented as follows:(19)∂2ℓω|X∂β12=−n1β12−α11+α12+1∑i=1Jυixiβ1logxi21+xiβ12−S1−n1α11+α12xJβ1logxJ21+xJβ12,∂2ℓω|X∂β22=−n2β22−α21+α22+1∑i=1J1−υixiβ2logxi21+xiβ22−S1−n1α21+α22xJβ2logxJ21+xJβ22,∂2ℓω|X∂αkj2=−mkjαkj2|k,j=1,2,∂2ℓω|X∂αkj∂αil=0,For eachkj≠il,∂2ℓω|X∂β1∂α1j=∂2ℓω|X∂α1j∂β1=−∑i=1Jυixiβ1logxi1+xiβ1−S1−n1xJβ1logxJ1+xJβ1,j=1,2,∂2ℓω|X∂β2∂α2j=∂2ℓω|X∂α2j∂β2=−∑i=1J1−υixiβ2logxi1+xiβ2−S2−n2xJβ2logxJ1+xJβ2,j=1,2,∂2ℓω|X∂β1∂α2j=∂2ℓω|X∂β2∂α1j=∂2ℓω|X∂α2j∂β1=∂2ℓω|X∂α1j∂β2=∂2ℓω|X∂β1∂β2=∂2ℓω|X∂β2∂β1=0.Suppose that the fisher information matrix is defined byΨβ1,β2,α11,α12,α21,α22, where(20)Ψβ1,β2,α11,α12,α21,α22=−E∂2ℓω|X∂ωi∂ωl,i,l=1,2,…,6,where ω=β1,β2,α11,α12,α21,α22 be the model parameters. Equation (19) has shown that the expectations of the second derivative of the log likelihood function are more serious. Therefore, we applied the approximate information matrix Ψ^0β^1,β^2,α^11,α^12,α^21,α^22 defined by(21)Ψ^0β^1,β^2,α^11,α^12,α^21,α^22=∂2ℓω|X∂ωi∂ωlβ^1,β^2,α^11,α^12,α^21,α^22,i,l=1,2,…,6.Therefore,Ψ^0−1β^1,β^2,α^11,α^12,α^21,α^22 exists with nonzero values of the elements of diagonal. Under normal properties of β^1,β^2,α^11,α^12,α^21,α^22, the approximate 1−2θ% confidence intervals of the parameters β1,β2,α11,α12,α21, and α22 are given by(22)β^1∓zθϵ11,β^2∓zθϵ22,α^11∓zθϵ33,α^12∓zθϵ44,α^21∓zθϵ55,α^22∓zθϵ66,where ϵil is the element of diagonal of the invariance approximate information matrix Ψ^0−1β^1,β^2,α^11,α^12,α^21,α^22 with significant level θ.


## 4. Bootstrap Confidence Intervals

In this section, we discussed a bootstrap technique in statistical inference problem about parameters estimation. This technique is a commonly resembling method not only in parameter estimation but also used to estimate bias and variance of an estimator or calibrate hypothesis tests. The bootstrap technique is defined in parametric and nonparametric methods (see Davison and Hinkley [30] and Efron and Tibshirani [31]). Therefore, we adopted parametric bootstrap technique to build two different confidence intervals, percentile bootstrap technique, and bootstrap-t technique. For more details, see Efron [32] and Hall [33]. The following algorithms are used to describe the procedure that is used to build different two bootstrap confidence intervals:(1)
Under consideration that the original observed Type-I JCRSX=X1,υ1,δ1,X2,υ2,δ2,…,XJ,υJ,δJ, the estimates are obtained and given by ω^=β^1,β^2,α^11,α^12,α^21,α^22.(2)
For givenω^ and integer values of N, S1S2, and time τ, generate a sample of size S1 from Burr XII distribution with shape parameters β^1 and α^11+α^12 and a sample of size S2 from Burr XII distribution with shape parameters β^2 and α^21+α^22. The τ-bootstrap Type-I JCRS is obtained from the generated joint sample as a small J satisfies that XJ<τ denoted by X=X1∗,υ1,δ1,X2∗,υ2,δ2,…,XJ∗,υJ,δJ.(3)
From Step 2, the two numbersn1∗ and n2∗ (number of failure taken from line η1 and η2, respectively) are obtained.(4)
The four numbersm1j∗ and m2j∗, j=1,2, are randomly generated from binomial distribution with size J−n3−k∗ and probability α^kj/α^k1+α^k2, k,j=1,2.(5)
The bootstrap estimate sampleω^∗=β^1∗,β^2∗,α^11∗,α^12∗,α^21∗,α^22∗ is obtained.(6)
Repeat Steps 2 to 5M times.(7)
The valuesβ^1i∗,β^2i∗,α^11i∗,α^12i∗,α^21i∗,α^22i∗, i=1,2,…,M, are arranged in ascending order to obtain ω˜∗=β^1i∗,β^2i∗,α^11i∗,α^12i∗,α^21i∗,α^22i∗.


### 4.1. Percentile Bootstrap Confidence Interval (PBCI)

Suppose that the ordered sample described by distributionΦx=Pω˜l∗≤x, l=1,2,3,4,5,6, be cumulative distribution function of ω˜l∗, where ω˜1∗ mean β^1∗ and others. So, the point bootstrap estimate is defined by(23)ω^l∗=1M∑i=1Mω˜li∗.Also, the1001−2θ% PBCIs are given by(24)ω˜lbootθ∗,ω˜lboot1−θ∗,where ω˜lboot∗=Φ−1x.


### 4.2. Bootstrap-t Confidence Interval (PTCI)

From the order sampleω˜∗=β^1i∗,β^2i∗,α^11i∗,α^12i∗,α^21i∗,α^22i∗, we built the order statistics values Φl∗1<Φl∗2<⋯<Φl∗M, where(25)Φl∗i=ω˜1i∗−ω^1varω˜1i∗,i=1,2,…,M,l=1,2,3,4,5,6.The1001−2θ% PTCIs are given by(26)ω˜lboot−tθ∗,ω˜lboot−t1−θ∗,where the value ω˜lboot−t∗ is given by(27)ω˜lboot−t∗=ω^l∗+Varω^lΦ−1x,and Φx=Pω˜l∗≤x be the cumulative distribution function of ω˜l∗.


## 4.1. Percentile Bootstrap Confidence Interval (PBCI)

Suppose that the ordered sample described by distributionΦx=Pω˜l∗≤x, l=1,2,3,4,5,6, be cumulative distribution function of ω˜l∗, where ω˜1∗ mean β^1∗ and others. So, the point bootstrap estimate is defined by(23)ω^l∗=1M∑i=1Mω˜li∗.Also, the1001−2θ% PBCIs are given by(24)ω˜lbootθ∗,ω˜lboot1−θ∗,where ω˜lboot∗=Φ−1x.


## 4.2. Bootstrap-t Confidence Interval (PTCI)

From the order sampleω˜∗=β^1i∗,β^2i∗,α^11i∗,α^12i∗,α^21i∗,α^22i∗, we built the order statistics values Φl∗1<Φl∗2<⋯<Φl∗M, where(25)Φl∗i=ω˜1i∗−ω^1varω˜1i∗,i=1,2,…,M,l=1,2,3,4,5,6.The1001−2θ% PTCIs are given by(26)ω˜lboot−tθ∗,ω˜lboot−t1−θ∗,where the value ω˜lboot−t∗ is given by(27)ω˜lboot−t∗=ω^l∗+Varω^lΦ−1x,and Φx=Pω˜l∗≤x be the cumulative distribution function of ω˜l∗.


## 5. Bayesian MCMC Estimation

In this section, we adopted Bayesian approach to estimate the model parameters under Type-I JCRS (see Ullah and Aslam [34]). So, we suppose that the prior information available about the parameters are independent Gamma prior distributions. Therefore, for parameters vectors ω=β1,β2,α11,α12,α21,α22, the prior information is defined by(28)Pi∗ωi=biaiΓaiωiai−1exp−biωi,ωi>0,ai,bi>0,i=1,2,3,4,5,6.And the corresponding density is defined by(29)P∗ω∝∏i=16ωiai−1exp−biωi.Therefore, the posterior distribution can be formulated by using (10) and (29) as follows:(30)Pω|X∝β1n1+a1−1β2n2+a1−1α11m11+a1−1α12m12+a1−1α21m21+a1−1α22m22+a1−1xJβ1−1xJβ2−1exp−b1β1−b2β2−b3α11−b4α12−b5α21−b6α22+β1−1∑i=1Jυilogxi−α11+α12+1∑i=1Jυilog1+xiβ1+β2−1∑i=1J1υilogxi−α21+α22+1∑i=1J1−υilog1+xiβ2−S1−n1α11+α12×log1+xJβ1−S2−n2α21+α22log1+xJβ2.The full conditional distributions are obtained from the joint posterior distribution (29), as follows:(31)P1β1|ω−1,X∝β1n1+a1−1exp−b1β1+β1−1∑i=1Jυilogxi−α11+α12+1×∑i=1Jυilog1+xiβ1−S1−n1α11+α12log1+xJβ1,(32)P2β2|ω−1,X∝β2n1+a2−1exp−b2β2+β2∑i=1J1−υilogxi−α21+α22+1×∑i=1J1−υilog1+xiβ2−S2−n2α21+α22log1+xJβ2,and the full conditional distributions of parameters αkj are gamma distributions given as(33)P3α1j|ω−1,X∝Gammam1j+aj+2,bj+2+∑i=1Jυilog1+xiβ1+S1−n1log1+xJβ1,j=1,2,(34)P4α2j|ω−1,X∝Gammam2j+aj+4,bj+4+∑i=1J1−υilog1+xiβ2+S2−n2log1+xJβ2,j=1,2,where the conditional value ωi|ω−1 means that the conditional i-th parameter for given the parameter vector ω without the i-th parameter ωi. The point and interval estimate of model parameters under MCMC methods depend on the forms of full conditional distributions and the subclass of MCMC that can be applied. Therefore, full conditional distribution given by (31) to (34) has shown that we can use the algorithms of Gibbs and generally Metropolis Hasting (MH) under Gibbs (for more details, see [35]) described in Algorithm 1.Algorithm 1: MH under Gibbs algorithms.
(1)
Putξ=1 and ω0=β^1,β^2,α^11,α^12,α^21,α^22 as initial values(2)
The parametersαkj are generated from Gamma distributions (32) and (33)(3)
With normal proposal distribution with the accepted rejection method with meanβkξ−1 and variances ϵi, generate βiξ,i=1,2(4)
Putξ=ξ+1(5)
Steps 2 to 4 are repeatedM times and report the vector ωξ=β1ξ,β2ξ,α11ξ,α12ξ,α21ξ,α22ξThe problem of generation under the MCMC method needs to determine the number of iteration needed to reach stationary distribution (burn-in) which is defined byM∗. Therefore, the point estimate is reduced to(35)ω˜lB=EPωl|X=1M−M∗∑i=M∗+1Mωli,l=1,2,3,4,5,6,and the corresponding variance is reduced to(36)V^ωl|X=1M−M∗∑i=M∗+1Mωli−ω˜lB2.Also,1001−2θ% credible intervals are obtained from ordered vectors given by(37)ωlθM−M∗,ωl1−θM−M∗.


## 6. Real Data Analysis

In this section, we analyzed a real data set presented by Hoel [36] to present the failure times and the corresponding cause of failure for two groups of strain male mice under laboratory experiment received a radiation dose of 300r at an age of 5-6 weeks. The life data are presented in Table 1, and let η1 be considered as the first group which lived in a conventional laboratory environment, but η2 be the second group lived in a germ-free environment. The data are classified into two causes of failure: thymic lymphoma with reticulum cell sarcoma as the first cause of death (failure) and the second cause is presented by other causes of death (failure); more details are presented by Koley and Kundu [37]. For simplicity, the data are divided by 1000.Table 1
Time-to-failure of male mice which received a radiation dose at age 5-6 weeks.
η1Thymiclymphoma159189191198200207220235245250256261265266280343356383403414428432Reticulumcell sarcoma317318399495525536549552554557558571586594596605612621628631636643647648649661663666670695697700705712713738748753Other cases40425162163179206222228249252282324333341366385407420431441461462482517517524564567586619620621622647651686761763η2Thymiclymphoma158192193194195202212215229230237240244247259300301321337415434444485496529537624707800Reticulumcell sarcoma430590606638655679691693696747752760778821986Other cases13624625537642156561661765265565866066267568173473673775776977780080782585585786486887087388289591093494210151019Therefore, the observed Type-I JCRS is taken from two lines of productionη1 and η2 under censoring scheme N=181,S1=99,S2=82, and τ=0.50 and is reported in Table 2. The data given in Table 2 show that n1,n2=50,30, m11,m12,m21,m22=26,24,25,5, and J=80. Figure 1 shows the joint profile log-likelihood function (18), and the value (2, 2) is a suitable initial value needed in the iteration method. The point estimate under ML, bootstrap, and Bayes estimators for noninformative prior information (mean ai=bi=0.0001, i=1,2,3,4,5,6) is reported in Table 3. And, the corresponding 95% approximate ML, two bootstrap confidence (Bootstrap-p and Bootstrap-t), and credibly intervals are, respectively, reported in Table 4. The generation results of full conditional distribution as a generation from posterior distribution and its convergence for Bayesian approach under MCMC methods are described in Figures 2 to 7 which have shown the quality of posterior generation.Table 2
Type-I JCRS from heal data withτ=0.5.
Data0.0400.0420.0510.0620.1360.1580.1590.1630.1790.1890.1910.1920.1930.1940.1950.1980.2000.2020.2060.2070.2120.2150.2200.2220.2280.2290.2300.2350.2370.2400.2440.2450.2460.2470.2490.2500.2520.2550.2560.2590.2610.2650.2660.2800.2820.3000.3010.3170.3180.3210.3240.3330.3370.3410.3430.3560.3660.3760.3830.3850.3990.4030.4070.4140.4150.420.4210.4280.4300.4310.4320.4340.4410.4440.4610.4620.4820.4850.4950.496(η1 or η2)11110011111000011011001110010001001110101111100110110111101111110101011010111010(δ1 or δ2)22222112211111111121111221111111212122111111211111221211221211211221121121222111Figure 1
The profile loglikelihood ofα1 and α2.Table 3
The point ML, bootstrap, and Bayes estimate.
Methodβ1β2α11α12α21α22⋅ML1.864 91.947 31.495 71.380 61.691 10.338 2⋅Boot2.001 31.854 11.625 41.564 21.745 20.625 4⋅B−MCMC1.751 41.821 71.303 31.202 51.456 40.291 0Table 4
95% ML, bootstrap, and Bayes interval estimate.
Pa.ACILengthBoo-pBoot-tCILengthβ1(1.577, 2.153)0.577(1.474, 2.65)1.180(1.452, 2.274)0.822(1.343, 2.232)0.889β2(1.665, 2.064)0.397(1.212, 2.845)1.633(1.275, 2.414)1.138(1.273 7, 2.448)1.174α11(1.066, 1.926)0.860(0.422, 2.854)2.433(0.748, 2.184)1.436(0.754, 2.147)1.393α12(0.955, 1.806)0.851(0.425, 2.321)1.396(0.692, 2.066)1.374(0.675, 1.969)1.294 06α21(1.298, 2.084)0.786(0.215, 2.965)1.896(0.746, 2.624)1.878(0.757, 2.521)1.765α22(0.062, 0.615)0.553(0.001, 0.966)0.964(0.020, 0.659)0.639(0.085, 0.645)0.559Figure 2
Simulation MCMC generated number/histogram of the parameterα1.Figure 3
Simulation MCMC generated number/histogram of the parameterα2.Figure 4
Simulation MCMC generated number/histogram of the parameterβ11.Figure 5
Simulation MCMC generated number/histogram of the parameterβ12.Figure 6
Simulation MCMC generated number/histogram of the parameterβ21.Figure 7
Simulation MCMC generated number/histogram of the parameterβ21.


## 7. Simulation Studies

The proposed model and its theoretical results in section are assessed and compared through the Monte Carlo study. So, we built this study to measure the effect of changing each of random sample sizeN=S1+S2, the test time τ, and parameters values. The values of sample size and the corresponding test time used in simulation study are reported in Tables 5 to 8. However, for the parameter values choosing, we used two sets, ω=2.0,1.2,1.3,1.8,2.0,2.0 and {1.0, 2.0, 3.0, 2.0, 2.5, 1.0}. In our studying, we generate 1000 simulated data sets. The prior parameters are selected to satisfy the property that Eωi≃ai/bi and information presented with two cases noninformative defined by P0∗ and informative prior P1∗. The informative prior P1∗ is taken to be (a, b) = {(3, 0.8), (2, 1.5), (2, 2), (2, 1), (3, 1.5), (4, 2)} for the first selected parameter values. And the informative prior information for the second selection of the parameters values is (a, b) = {(2, 2), (2, 2), (3, 1.2), (4, 2), (4, 1.5), (1, 1)}. Also, through this problem, mean estimate (ME) and the corresponding mean squared error (MSE) are used to measure the point estimate. And, mean interval length (MIL) and probability coverage (PC) are used to measure interval estimate. The Monte Carlo study is done with respect to Algorithm 2.Algorithm 2: General steps used to generate Type-I joint competing risk samples and the corresponding estimate (see Almarashi et al. [15]).
(1)
Two samples of sizeS1 and S2 are generated form Burr XII distribution with parameters βk and αk1+αk2,k=1,2, respectively. Hence, the joint sample of size N=S1+S2 is generated.(2)
For givenτ, the Type-I JCRS and its size J are determined.(3)
The integersn1 and n2 are computed from the Type-I JCRS.(4)
The random integersmkj are generated from binomial distributions.(5)
Steps 1 to 4 are repeated 1000 times to obtain 1000 Type-I JCRS.(6)
The MLE, bootstrap, and Bayes point and intervals estimates are computed for each sample.(7)
The values of each ME, MSEs, MILs, and PCs are computed and reported in Tables5–8.Table 5
The ME and MSEs of ML, boot, and Bayes methods underω=2.0,1.2,1.3,1.8,2.0,2.0.
τ,S1,S2β1β2α11α12α21α22MEMSEMEMSEMEMSEMEMSEMEMSEMEMSE(0.2, 30, 30)ML2.5420.4541.4230.4231.5240.4212.3520.6322.4570.4892.3970.500Boot2.5770.4811.6250.4881.5990.5022.3870.6892.4990.5322.4210.552BayesP02.5370.4521.4140.4111.5170.4112.3410.6182.4420.4812.3870.492BayesP12.4250.3551.3170.3181.4270.3572.2400.5182.3140.3922.3010.380(0.2, 50, 30)ML2.5210.4281.4190.4421.4810.3952.3290.6012.4490.4832.3900.492Boot2.5550.4521.6270.4931.5480.4812.3650.6612.4810.5282.4170.553BayesP02.5190.4241.4180.4211.4780.3922.3180.5852.4450.4842.3790.487BayesP12.4070.3211.3130.3131.4030.3312.2150.4812.3070.3872.3030.371(0.2, 40, 60)ML2.5250.4311.3790.4041.4880.3912.3250.5982.4070.4522.3610.459Boot2.5510.4491.6000.4581.5390.4832.3540.6632.4380.4902.2750.511BayesP02.5220.4181.3780.3811.4790.3882.3210.5792.4090.4492.3410.451BayesP12.4030.3171.2810.2791.2970.3272.2190.4802.2780.3512.2690.339(0.2, 80, 80)ML2.5000.4001.3620.2921.4290.3582.3000.5632.2980.4472.3540.451Boot2.5120.4111.5910.4411.5120.4482.3180.6182.4300.4832.2700.501BayesP02.4770.2811.3710.3731.4410.3392.2920.5432.3970.4322.3370.444BayesP12.2750.2821.2750.2701.2630.3002.1890.4392.2670.3382.2540.328(0.8, 30, 30)ML2.5070.4071.3710.2961.4410.3652.3080.5712.3010.4522.3620.457Boot2.5180.4181.5980.4471.5180.4572.3150.6242.4470.4892.2790.514BayesP02.4810.2931.3770.3751.4530.3512.2990.5552.4050.4382.3470.457BayesP12.2790.2851.2810.2771.2690.3082.1610.4492.2780.3452.2630.341(0.8, 50, 30)ML2.4810.3801.3740.2921.4070.3282.2740.5382.3070.4502.3600.449Boot2.4890.3851.5930.4411.4750.4142.2810.5712.4410.4912.2710.510BayesP02.4550.2611.3720.3701.4110.3042.2540.5142.4010.4332.3420.458BayesP12.2440.2851.2840.2791.2190.1722.1470.4042.2810.3422.2600.338(0.8, 40, 60)ML2.4870.3851.3310.2631.4110.3312.2740.5412.2790.4182.3280.411Boot2.4930.3871.4790.4041.4790.4222.2890.5782.4080.4582.2370.471BayesP02.4590.2641.3380.3181.4180.3142.2600.5252.3690.4042.3150.411BayesP12.2480.2891.2240.2321.2210.1832.1510.4132.2340.3002.2180.300(0.8, 80, 80)ML2.4150.3411.3110.2451.3900.2902.2150.5022.2550.4022.3020.292Boot2.4250.3591.4520.3811.4620.3822.2270.5142.4010.4412.2180.449BayesP02.4110.2411.3140.3011.3900.2712.2110.5002.3540.2922.3010.395BayesP12.2070.2421.2000.2141.2000.1442.1140.351 32.2180.2712.2020.281Table 6
The MILs and CPs of ML, boot, and Bayes methods underω=2.0,1.2,1.3,1.8,2.0,2.0.
τ,S1,S2β1β2α11α12α21α22MILCPMILCPMILCPMILCPMILCPMILCP(0.2, 30, 30)ML4.1420.853.2110.873.5120.884.2150.895.4130.885.1240.88Boot4.3140.883.4150.883.6880.894.3890.895.5980.895.3110.88BayesP04.1010.893.1780.893.4810.904.1820.905.3890.895.0470.90BayesP13.2450.913.0010.903.2140.914.0070.905.2170.904.8740.91(0.2, 50, 30)ML3.8500.902.9010.893.0120.903.8410.905.2110.905.1090.90Boot3.9500.903.0800.903.1300.904.0070.895.4170.895.2140.89BayesP03.8110.912.9200.903.0110.923.7000.925.1980.895.0010.90BayesP13.1240.932.7100.912.9100.963.5070.944.9870.924.7800.94(0.2, 40, 60)ML3.8990.892.7250.893.0250.903.8710.915.0030.904.9520.91Boot3.9810.902.8950.913.1420.914.0190.895.2740.925.0010.91BayesP03.8610.922.7010.903.0190.923.7220.905.0000.914.8900.92BayesP13.1900.912.5030.932.9270.903.5310.914.7110.934.5200.91(0.2, 80, 80)ML3.6010.902.6100.912.9110.913.5900.914.8020.914.8150.91Boot3.6900.922.7810.943.0000.933.8110.905.0030.925.9120.91BayesP03.5700.922.5900.902.8900.903.5300.924.7790.904.7150.91BayesP12.8540.932.4010.942.7110.943.2240.934.4900.924.4120.95(0.8, 30, 30)ML3.6540.892.6800.902.9780.913.6100.894.8750.904.8640.90Boot3.6970.902.7750.903.0690.903.8820.905.0690.925.9490.90BayesP03.6300.902.6400.912.9450.903.5910.904.8190.924.7600.93BayesP12.9150.902.5010.922.7740.913.3050.924.5000.924.4450.92(0.8, 50, 30)ML3.4180.912.6710.902.7600.913.4010.914.8790.894.8710.90Boot3.4980.902.7540.932.8790.933.6700.925.0850.915.9390.90BayesP03.4010.912.6210.912.8470.923.4020.944.8040.914.7480.91BayesP12.7210.922.4910.902.5760.943.1150.944.5030.944.4450.93(0.8, 40, 60)ML3.4410.892.4510.912.7720.913.4240.904.6100.964.6230.90Boot3.5140.902.5240.972.8810.903.6790.914.8500.925.7900.91BayesP03.4220.922.4050.922.8420.913.4140.904.5790.944.6240.91BayesP12.7380.922.2130.922.4950.923.0080.914.2800.954.2150.94(0.8, 80, 80)ML3.1500.932.1780.922.6840.923.3140.924.3900.904.3980.93Boot3.2410.922.2900.902.7010.923.4500.914.5120.935.5880.90BayesP03.1100.932.1540.942.6290.923.3750.934.3700.944.4010.93BayesP12.4150.952.0010.952.4780.953.0000.934.0030.934.0050.96Table 7
The ME and MSEs of ML, boot, and Bayes methods underω=1.0,2.0,3.0,2.0,2.5,1.0.
τ,S1,S2β1β2α11α12α21α22MEMSEMEMSEMEMSEMEMSEMEMSEMEMSE(0.1, 30, 30)ML1.2340.2342.4210.4213.3420.6452.3240.4022.8420.4951.3210.255Boot1.3350.2382.5420.4263.4520.6482.4260.4052.8900.4981.4350.291BayesP01.2290.2332.4180.4203.3020.6442.3140.4002.8340.4931.3130.243BayesP11.1980.2242.3700.4113.2010.6342.2800.3902.7900.4841.2800.242(0.1, 50, 30)ML1.2170.2172.4080.4033.2640.6272.3050.3872.8290.4771.3040.237Boot1.3190.2212.5280.4093.4010.6212.4070.3912.8750.4811.4210.278BayesP01.2080.2152.4040.4013.2550.6272.2930.3852.8170.4721.3000.225BayesP11.1810.2092.3670.3823.1850.6132.2610.3692.7770.4681.2660.222(0.1, 40, 60)ML1.2230.2222.3650.3673.2720.6312.3140.3952.7800.4411.2850.209Boot1.3250.2272.4510.3713.4150.6272.4120.3972.8310.4491.3700.251BayesP01.2140.2182.3610.3623.2670.6352.2990.3882.7770.4471.2750.201BayesP11.1930.2132.3220.3483.1920.6212.2740.3742.7310.4151.2140.191(0.1, 80, 80)ML1.1750.1872.3140.3283.2310.5612.2720.3512.7350.4071.2510.172Boot1.2860.1922.4070.3473.3700.5882.3500.3622.8000.4151.3610.199BayesP01.2690.1792.3210.3273.2290.5642.2640.3472.7220.4011.2540.162BayesP11.1440.1522.2790.3013.1510.5412.2170.3222.6800.3621.1820.127(0.5, 30, 30)ML1.2110.2142.4070.4013.3240.6152.2850.2842.8150.4761.3090.241Boot1.3100.2172.5310.4093.4280.6192.4040.2922.8670.4811.4240.279BayesP01.2030.2122.4000.3973.2870.6172.2890.2712.8070.4821.3040.231BayesP11.1750.2052.3560.3943.1800.6012.2600.3622.7660.4611.2710.228(0.5, 50, 30)ML1.2000.2002.3970.3853.2440.6002.2840.3652.8010.4551.2920.222Boot1.3020.2042.5100.3973.3810.5932.3910.3732.8490.4631.4090.266BayesP01.1850.1972.3890.3873.2370.6012.2730.3672.7900.4541.2910.210BayesP11.1600.1922.3510.3623.1610.5862.2440.3512.7510.4491.2520.214(0.5, 40, 60)ML1.2070.2012.3470.3493.2510.6032.2920.3762.7510.4241.2520.189Boot1.3030.2042.4380.3553.4000.6012.3940.3802.8020.4301.3580.238BayesP01.1910.2002.3470.3413.2440.6072.2810.3692.7480.4181.2630.187BayesP11.1810.1922.3070.3293.1750.5902.2550.3592.7030.4001.2000.177(0.5, 80, 80)ML1.1620.1702.3000.3093.2140.5342.2490.3352.7080.3891.2390.160Boot1.2710.1732.3940.3283.3520.5592.3290.3472.7710.4011.3500.184BayesP01.2520.1582.3020.3073.2070.5372.2410.3222.7000.3811.2410.151BayesP11.1290.1342.2620.2823.1330.5152.2000.3012.6530.3411.1700.114Table 8
The MILs and CPs of ML, boot, and Bayes methods underω=1.0,2.0,3.0,2.0,2.5,1.0.
τ,S1,S2β1β2α11α12α21α22MILCPMILCPMILCPMILCPMILCPMILCP(0.1, 30, 30)ML2.5140.884.5210.895.9850.873.9850.905.2340.902.4510.90Boot2.6740.894.6540.896.1240.894.2310.895.3850.902.5620.91BayesP02.4850.904.4920.905.9410.893.9370.905.2010.902.4140.90BayesP12.3500.904.3200.915.7520.913.8130.905.0250.902.2850.91(0.1, 50, 30)ML2.4650.904.4720.905.9440.903.9410.905.1920.912.4000.90Boot2.6270.914.6010.896.0510.914.1880.905.3410.902.5230.90BayesP02.4410.904.4510.915.9000.913.9000.925.1540.912.3660.94BayesP12.3040.934.2750.925.7070.923.7620.924.9550.932.2410.92(0.1, 40, 60)ML2.4540.904.4390.905.9360.903.9390.905.1550.912.3600.92Boot2.6310.894.5690.926.0610.914.1750.925.3110.932.4810.91BayesP02.4470.904.4180.915.9030.923.9090.905.1190.912.3280.93BayesP12.3140.904.2450.945.7150.913.7690.934.9210.942.2070.96(0.1, 80, 80)ML2.4070.904.4010.925.9020.913.9030.925.1110.942.3600.92Boot2.5820.924.5150.936.0240.924.1320.925.2710.932.4810.90BayesP02.4000.944.3750.915.8710.923.8610.925.0620.932.3280.94BayesP12.2670.954.2080.965.6740.943.7150.944.8740.942.1870.95(0.5, 30, 30)ML2.4850.894.5020.895.9550.893.9610.915.2050.912.4360.92Boot2.6410.904.6350.896.1010.894.2090.905.3440.912.5440.91BayesP02.4490.904.4740.915.9150.903.9150.905.1450.902.4000.92BayesP12.3240.904.3010.915.7270.913.8000.915.0000.922.2710.91(0.5, 50, 30)ML2.4410.914.4560.905.9190.903.9180.925.1510.912.3780.90Boot2.6000.914.5800.906.0240.924.1610.925.3020.922.5020.92BayesP02.4150.914.4330.915.8750.913.8820.925.1150.912.3410.92BayesP12.2740.934.2570.935.6910.923.7440.924.9120.922.2210.92(0.5, 40, 60)ML2.4270.914.4180.905.9140.923.9250.925.1140.912.3390.92Boot2.6070.904.5500.926.0320.914.1510.925.3110.932.4600.94BayesP02.4210.904.4010.915.2740.933.8920.905.0660.922.3090.93BayesP12.2880.924.2270.935.8940.913.7510.934.8690.942.1910.96(0.5, 80, 80)ML2.3810.934.3840.925.8770.923.8770.935.0700.922.3390.92Boot2.5550.924.5000.925.9820.944.1140.925.2390.932.4550.94BayesP02.2740.924.3620.915.8500.923.8510.925.0300.952.3070.94BayesP12.2310.954.1890.945.6440.953.7030.924.8350.942.1790.94


## 8. Conclusions

Recently, the joint censoring scheme is more widely used in a comparative life testing specially for products coming from different lines of production. The problem of comparative life testes under different causes of failure has been discussed recently under the joint censoring scheme of competing risks exponential lifetime model by Almarashi et al. [15]. In this paper, we adopted this problem when units or individual is distributed with Burr XII distributions. The unknown model parameters are estimated with classical methods (ML and bootstrap) and Bayes method with noninformative and informative prior. Numerical computation is exposed with real data analysis and Monto Carlo simulation study to assess and discuss the developed results. The numerical result discusses changing of sample size, test time, and available information. Therefore, we observed the following points:(1)
The proposed model under Type-I JCRS serves well for all choice of censoring schemes and parameters choices(2)
The Bayes estimation under noninformative priorP0 is more close to maximum likelihood estimation(3)
The informative priorsP1 serve better than noninformative prior and maximum likelihood estimations(4)
The increasing effect of sample sizeS1+S2 reduces the MSE and MIL(5)
The large value of test timeτ serves well than small value of τ



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*Source: 9553617-2021-12-24.xml*