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

(e 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. (is 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).(e life Lomax products’ model formulation from two different lines of production with BJPHCS is discussed. (e 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. (e 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.


Introduction
e 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. e 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. e 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][2][3][4]. e joint cases of type-I and type-II or progressive type-I and progressive type-II are called hybrid censoring schemes.
Suppose n 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 � {R 1, R 2 , . . ., R m } such as n � m + m i�1 R i are determined. At each failure time T i , R i 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, T m ), but, in a progressive hybrid type-II censoring scheme, the experiment is terminated at max (τ, T m ).
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. erefore, suppose two lines of production A 1 and A 2 produced the same product under the same conditions. Suppose two independent samples κ 1 and κ 1 are selected from the lines A 1 and A 2 , respectively, to be put simultaneously under life testing experiment. In the literature, joint censoring has been discussed in [5,6]. e classical estimation is discussed in [7,8]. Bayes estimation is presented by [9] and recently by [10][11][12].
For jointly progressive hybrid type-I censoring scheme, the total sample with size (κ 1 +κ 2 ) which is collected from the lines A 1 and A 2 , where κ 1 from A 1 and κ 2 from A 2 , are put under the test. en, the number of failure units m, the censoring scheme R � R 1 , R 2 , . . . , R m , and the ideal test time τ is given at the prior of the experiment. e failure time and the corresponding unit type (from A 1 or A 2 ), say (T i , w i ), is record for i � 1, 2, . . . , m, where w i � 1 { , 0} is denoted to unit type, from line A 1 or line A 2 , respectively. en, at any step of the experiment, when the failure time (T i , w i ) is observed, R i survival units are randomly removed from the experiment. e experiment is terminated when min (τ, T m ) is recorded and the random sample, say However, if we consider the jointly progressive hybrid type-II censoring scheme, the experiment is terminated when the max (τ, T m ) is observed and the random sample, say 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].
is 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, A 1 and A 2 , have the same kind of products under the same facility and the sample of size (κ 1 + κ 2 ) with size κ 1 from A 1 and κ 2 from A 2 are put under life testing. Also, the prior integer m is denoted to the number of failed units, and censoring scheme R � R 1, , R 2 , . . . , R m− 1 is determined to satisfy m + m− 1 i�1 R i < min(κ 1 , κ 2 ). If the first failure T 1 is observed from the line A 1 , then R 1 and (R 1 + 1) survival units are removed from the sample κ 1 and κ 2 , respectively. Also, if the second failure T 2 is observed from the line A 2 , then (R 2 + 1) and R 2 survival units are removed from the sample (κ 1 − 1 − R 1 ) and (κ 2 − 1 − R 1 ), respectively. en, the test is continual until mth failure is reached. If the mth failure is from the line A 1 , then the remaining survival units drown from the test. en, the observed balanced joint progressive type-II censoring sample is given by t � (t 1 , w 1 ), (t 2 , w 2 ), . . . , (t m , w m ) .
is paper aims to use the progressive hybrid type-I censoring scheme under a balanced joint technique to present the BJPHCS. e 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. e 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.
e 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. e paper is constructed with the following sections. e concepts and model of the formulation are built in Section 2. e model parameters are estimated with maximum likelihood estimation in Section 3. e 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.

The Model
Suppose the product comes from different two lines A 1 and A 2 of production under the same conditions. And, let the sample of size (κ 1 + κ 2 ) be randomly selected with κ 1 drawn from the line A 1 and κ 2 drawn from the line A 2 . 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 � R 1, R 2 , . . . , R m to satisfy m + m− 1 i�1 R i < min(κ 1 , κ 2 ). When the experiment is beginning, the time to failure and its type are recorded, which means from line A 1 or line A 2 . If the first failure T 1 is recorded from the line A 1 , then R 1 and (R 1 + 1) survival units are removed from the sample κ 1 and κ 2 , respectively. Also, if the second failure T 2 is observed from the line A 2 , then (R 2 + 1) and R 2 survival units are removed from the sample (κ 1 − R 1 − 1) and (κ 2 − R 1 − 1), respectively. en, the experiment is continual until the min (τ, T m ) is observed. If T m < τ, the results are similar to balanced joint progressive type-II censoring [13], but, if T m > τ, the experiment is removed at τ, where T J < τ < T J+1 . If the Jth failure is from the line A 1 , then the remaining ( en, the observed BJPHC sample is given by t � (t 1 , w 1 ), (t 2 , w 2 ), . . . , (t J , w J )}. Figure 1 shows the plane of BJPHCS. Suppose κ 1 units have the identical independent distributed (i.i.d.) random lifetimes X 1 , X 2 , . . . , X κ 1 and κ 2 units have the i.i.d. random lifetimes Y 1 , Y 2 , . . . , Y κ 2 . e two samples are distributed with distribution with probability density functions (PDFs), and the cumulative distribution functions (CDFs) are given by f l (.) and F l (.), l � 1, 2.
en, the ordered sample where J � J 1 + J 2 and J 1 is the number of units fails from the line A 1 and J 2 is the number of units fails from the line A 2 .
e observed BJPHCS censoring sample t � (t 1 , w 1 ), (t 2 , w 2 ), . . . , (t J , w J ) in which w i is 1 or 0 values and depends on line A 1 or A 2 , respectively, and under observed sample t � (t 1 , w 1 ), (t 2 , w 2 ), . . . , (t J , w J ) , the likelihood function is given by 2 Complexity where τ * � min(τ, T m ), Φ is the parameters vector, and S j (.) and h j (.), j � 1, 2, are denoted as reliability and hazard rate functions, respectively. Different lines A 1 and A 2 of production with units have PDFs, and CDFs follow Lomax lifetime distribution defined with PDFs and given by and CDFs are given by e reliability and hazard rate functions of Lomax lifetime distributions is also given by e 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. e failure time of Lomax lifetime distribution under different risks is studied by [18] and recently by [19].

Maximum Likelihood Estimation
For the given joint sample t � � {(t 1 , w 1 ), (t 2 , w 2 ), . . ., (t J , w J )} and Lomax distribution given by (2)-(5), the joint likelihood function (1) is reduced to where e joint likelihood function is defined by (6) after the natural logarithms are reduced to e 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.

Bayes Estimation
e 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: where Φ presents the parameters' vector. e joint prior density is given by 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 where Q � Φ Π * (Φ)L(Φ | t)dΦ. en, the Bayes estimators for the function ζ(Φ) under squared error loss function (SEL) is given by Hence, the Bayes estimators have two integrables, which need approximation methods such as numerical integration or Lindley approximation. e 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 e full conditional distributions of the parameters are given by en, the posterior distribution is reduced to two conditional gamma functions and two distributions more similar to normal distribution. en, the posterior distribution (25) and conditional posterior distributions (26) to 6 Complexity (29) have shown that MCMC methods are more suitable with more general Metropolis-Hastings (MH) under Gibbs algorithms, see [22] described as follows.

MH under Gibbs algorithm
Step 1: for given initial vectors Step 2: generate two values α (I) 1 and α (I) 2 from conditional gamma distribution given by (26)  Step 5: change I to be I + 1 Step 6: steps (2) to (5) are followed to get iteration procedure to M times Step 7: to reach the stationary state, we need iteration number M * which is known by "burn-in;" then, the Bayes estimators is given by as well as the corresponding posterior variance of ζ(Φ) also, which is given by where ζ(Φ) is a function of the parameters vector may be any one of them.
Step 8: the obtaining empirical distribution of ζ(Φ) is obtained by the MCMC iteration after arranging its values ascending; then, the corresponding 100(1 − c)% credible interval of ζ(Φ) is given by (32)

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. e true parameters are chosen to satisfy that E(Φ i )≃(a i /b i ), 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 (κ 1 , κ 2 ) � (40, 40) and prior integers m � 25, censoring scheme R�{14, 0 (23) , 1}, and the ideal test time τ � 10. Hence, the randomly generated BJPHC data concerning to the algorithms are given by [24]. e generated data (t i , w i ) 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. e 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. en, the figures show the quality of the MCMC method in the Bayes method.

Simulation Studies.
is section has examined the best choice of sample size, censoring scheme, and ideal test time.
is 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 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 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.

Concluding Remarks
e commonly used problem in life products with different production lines is measuring the relative merits of the products competing for the duration. e 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.
e Bayes estimates are computed with squared error loss functions.
e Bayes estimators' explicit forms cannot be obtained, so the MCMC approach is impalement to get the s, interval length, and coverage probability (6) Censoring scheme with middle censoring performs better than other censoring schemes (7) e method of MCMC for approximation Bayesian estimate serves well, especially in dimensional cases

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors have no conflicts of interest regarding the publication of the paper.