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

Information Technology Department, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt Statistics Department, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Mathematics Department, Faculty of Science, Damanhour University, Egypt Statistics Department, Faculty of Commerce (Girl Branch), Al-Azhar University, Cairo, Egypt Department of Mathematics, College of Science, Taif University, Taif, Saudi Arabia Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt Department of Computers Science, Faculty of Computer and Information, Luxor University, Luxor, Egypt Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt


Introduction
Statistical inference for the life products needs to put some units of product under test to get more information about the life products. en, we design the life experiments to obtain the required data. Under consideration time and cost, data obtained may be complete or censored. e 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. e 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 by τ * 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 * , T m ). e 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 * , T m ) [1,2], but in type-II HCS, the experiment is removed from the test at the max (τ * , T m ) [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]. e 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, for n tested units, suppose prior integers s and m that satisfy 1 ≤ s < m ≤ n and prior ideal test time τ * ∈ (0, ∞). e three cases are considered. If T s < τ * , the test is terminated at min (τ * , T m ), and in other cases, if τ * < T s < T m , the test is terminated at T s , but if t m;n < τ * , the test is terminated at T m . e data in type-I GHCS satisfy the minimum number s needing for statistical inference, and the data are summarized as follows: (1) t � (t 1;n < t 1;n < . . . < t s;n ), if τ * < T s (2) t � (t 1;n < t 1;n < . . . < t k;n < . . . < t r;n ), if τ * > T s and t m;n > τ * (3) t � (t 1;n < t 1;n < . . . < t m;n ), if t m;n < τ * e scheme of the type-I GHCS can be formulated with the schematic diagram described by Figure 1.
Type-II GHCS, for n tested units, suppose prior times τ * 1 and τ * 2 ∈ (0, ∞), such that τ * 1 < τ * 2 and integer m satisfies that 1 ≤ m ≤ n. e three cases are considered. If T m < τ * 1 , the test is terminated at τ * 1 , and in second case, if τ * 1 < T m < τ * 2 , the test is terminated at T m . If τ * 2 < T m , the test is terminated at τ * 2 . e data in type-II GHCS satisfy the maximum time τ * 2 , and the data are summarized as follows: e scheme of the type-II GHCS can be formulated with the schematic diagram described by Figure 2.
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 by Γ 1 and Γ 2 in competing duration and let two independent samples with sizes N 1 and N 2 , respectively. e joint sample of N 1 and N 2 is put under life testing. is experiment is restricted under consideration of time and cost to terminate after fixed time or number of failure. e data obtained from type of censoring are called joint samples discussed early in [7,8]. e 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. erefore, 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. e maximum likelihood and Bayes methods are applied for the parameters estimation.
e developed theoretical method assessed through the simulation study as well as illustration is reported with data analysis.
e study is planned as follows: the concept and model formulation are reported in Section 2. 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. e numerical computation is discussed through a simulation study in Section 6. Finally, some brief comments are reported in Section 7.

Model
Consider that the product comes from two different lines of production Γ 1 and Γ 2 that have the same facility. Suppose, independent two samples of sizes N 1 and N 2 selected from Γ 1 and Γ 2 have independent and identical distributed (i.i.d) lifetimes W 1 , W 2 , . . . , W N 1 and Z 1 , Z 2 , . . . , Z N 2 , respectively.
e independent samples distributed with populations have f j (.) probability density functions (PDFs) and F j (.) cumulative distribution functions (CDFs) for j � 1, 2.
e lifetime experiment begins with prior integers and ideal test time given by (s, m, τ * ). rough the experiment, the unit failure time, and its type, means from Γ 1 and Γ 2 are recorded. en, the experiment is continual until r th failure is observed; if τ * < T s , then r � s, but if τ * > T m , then r � m, and in other cases, s < r < m. e vector of ordered sample T � (T 2 , δ 1 ), (T 2 , δ 1 ), . . . , (T r , δ r ) from the sample W 1 , W 2 , . . . , W N 1r , Z 1 , Z 2 , . . . , Z N2r with r � N 1r + N 2r , 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, δ i , i � 1, 2, . . . , r, take the value 1 or 0 depending on failure unit from Γ 1 or Γ 2 , respectively, and the two integers m 1 and m 2 denote the number of units fails from lines Γ 1 and Γ 2 , respectively. e joint likelihood function from observed joint type-I GHSC t � {(t 2 , δ 1 ), (t 2 , δ 1 ), . . ., (t r , δ r )} and m 1 � r i�1 δ i , where and survival functions S j (.), j � 1, 2. Suppose that the lifetime T has two parameters Burr XII populations with PDFs given by and CDFs, survival functions S j (t), and failure rate function h i (t) are given by e function (4) shows that the parameter a i does not affect the shape of failure rate function h i (t). Burr XII has been applied in areas of quality control, reliability studies, duration, and failure time modeling, see for example [15].

Estimations under the Maximum
Likelihood Method e joint likelihood function (1) under Burr XII populations (3) and (4) and observed joint type-I GHS sample t � (t 2 , δ 1 ), (t 2 , δ 1 ), (t r , δ r ) is formed by Figure 1: Different types of type-I GHCS. Complexity e natural logarithms of (5) without a normalized constant reduce to 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: and are reduced to and Also, has presented that en, the nonlinear, equations (11) and (12) are reduced after replacing a 1 and a 2 from (8) and (9) to e two nonlinear equations presented by (13) and (14) present the likelihood equations of parameters b 1 and b 2 , 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 m 1 � 0 or m 2 � 0, the parameter values a 1 and b 1 or a 2 and b 2 , respectively, are difficult to obtain [16].

Approximate Interval Estimation.
e second partial derivatives of log-likelihood function (6) with respect to parameters vector ω � (a 1 , b 1 , a 2 .b 2 ) are given by e 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 Complexity 5 information matrix. Let η denotes the Fisher information matrix defined by where ω � (a 1 , b 1 , a 2 , b 2 ). en, the approximate information matrix of η denoted by η 0 is defined as Hence, under asymptotic normality distribution of and the vector (ϵ 11 , ϵ 22 , ϵ 33 , ϵ 44 ) presents the diagonal of the covariance matrix η −1 0 (a 1 , b 1 , a 2 , b 2 ), and the value z c is the percentile of the normal (0,1) with right-tail probability α.

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: where ω � (a 1 , b 1 , a 2 , b 2 ) is the model parameter. e joint prior distribution is given by (20) Generally, the posterior distribution from the model And the Bayes estimate under squared error loss function (SEL) of function Ω (a 1 , b 1 , a 2 , b 2 ) is given by e equation (22) has ratio of two integrals, generally cannot be obtained in a closed form. en, numerical approximation will be used to solve this problem. One way is called numerical integration, and other way used can be called Lindley approximate. e important method which has considerable attention in the last year called the MCMC method is discussed as follows.
e proportional form of joint posterior distribution (21) with joint prior distribution (20) and the likelihood function (6) is given by 6 Complexity e joint posterior distribution (23) reduced with the full-conditional probability distributions PDF's is given as follows: e 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. e 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.
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 by M * . en, the Bayes estimators are given by Also, the posterior variance of function Ω is given by (28) e credible interval is obtained with ordering the simulated vector Ω; then, 100(1 − 2α)% credible interval of function Ω is given by

Illustrative Example
In this section, we summarized a simulated dataset from two Burr XII populations to check the theoretical results discussed through the study. e real parameters value is selected to satisfy prior information, so that with (ρ 1 , ρ 2 , ρ 3 , ρ 4 ) � (2, 3, 4, 2) and (υ 1 , υ 2 , υ 3 , υ 4 ) � (2, 2.5, 2, 2), the real parameter values are selected to satisfy Bayes estimates as well as asymptotic confidence interval and credible interval are summarized in Table 2. e 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. en, we can test if stationary distribution is reached quickly starting from an arbitrary position. e 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.

Simulation Studies
e 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. en, 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. e simulation study is reported for different sample sizes (N 1 + N 2 ) 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 � (a 1 , b 1 , a 2  e prior parameters are selected to satisfy E Ψ * (ω i )≃(ρ i /υ i ). Hence, in our simulation study, we proposed different two cases from prior information; one of them is expressed to noninformative prior (prior 0 ), in which the posterior distribution is proportional with likelihood function. e second case is expressed to informative prior (prior 1 and prior 2 ). e prior 1 is (ρ 1 , ρ 2 , ρ 3 , ρ 4 ) � (1,3,4,5) and (υ 1 , υ 2 , υ 3 , υ 4 ) � (1, 2,1.5, 2). e prior 2 is (ρ 1 , ρ 2 , ρ 3 , ρ 4 ) � (1, 2.5, 3, 4) and (υ 1 , υ 2 , υ 3 , υ 4 ) � (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. e MCMC method is performed under 11000 chen with 1000 burn-in, and the results are reported in Tables 3-6.

Concluding Remarks
In the industrial field, the existing different lines of production have the same products under the same facility. e problem of measuring the relative merits of product in the competing duration has considerable attention in past view years.
is problem has been discussed in this study for products distributed with Burr XII lifetime distribution. is problem presented in parameters estimation forms with ML and Bayesian estimations under joint type-I GHCS. en, the developed method is assessed through the Monte Carlo simulation study. e results obtained from these studies show the following comments.             Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.