Statistical Inference of Chen Distribution Based on Two Progressive Type-II Censoring Schemes

: An inverse problem in practical scientific investigations is the process of computing unknown parameters from a set of observations where the observations are only recorded indirectly, such as monitoring and controlling quality in industrial process control. Linear regression can be thought of as linear inverse problems. In other words, the procedure of unknown estimation parameters can be expressed as an inverse problem. However, maximum likelihood provides an unstable solution, and the problem becomes more com-plicated if unknown parameters are estimated from different samples. Hence, researchers search for better estimates. We study two joint censoring schemes for lifetime products in industrial process monitoring. In practice, this type of data can be collected in fields such as the medical industry and industrial engineering. In this study, statistical inference for the Chen lifetime products is considered and analyzed to estimate underlying parameters. Maximum likelihood and Bayes’ rule are both studied for model parameters. The asymptotic distribution of maximum likelihood estimators and the empirical distributions obtained with Markov chain Monte Carlo algorithms are utilized to build the interval estimators. Theoretical results using tables and figures are adopted through simulation studies and verified in an analysis of the lifetime data. We briefly describe the performance of developed methods.


Introduction
Several types of monitoring data are available. One is the censoring scheme, which is a popular problem in life testing experiments. The oldest censoring projects are the so-called "type-I", and the other is "type-II". In practice, there are usually two random variables, i.e., time and the number of failures of items. This strategy of censoring projects shows how the examiner imagines the experiment based on a predetermined time. A random number of units is accounted for the first type-I of a censoring scheme, which means it may be assumed the exact time of stopping experiment. While the predetermined number of failure units and a random time in the type-II censoring scheme. In these two types of censoring schemes, companies cannot be removed from an experiment until the final stage or the number of units fail. This process allows the detection of some units that are defective after running the experiment. The mixture of these types of censoring schemes is the so-called hybrid censoring system [1]. To remove elements from the test at any stage of the trial is known as a progressive censoring scheme [2]. The topic of progressive censoring has developed in different scientific fields, and has attracted much attention in recent years. Several authors have studied this type of data [3,4]. There are different types of progressive censoring schemes. The idea of the progressive type-I censoring scheme is to test time τ and determine the number m of failure units, and suppose n independent elements are tested under the censoring scheme r = {r 1  . . , m. The lifetime products come from different production lines [5,6]. The exact likelihood inference using bootstrap algorithms was studied [7], as was the type-II progressive censoring scheme [8,9] and two censoring schemes [10]. Consider manufactured products that come from two production lines η 1 and η 2 under the same conditions. Assume two independent samples S 1 and S 2 are chosen from these lines for experimental testing. The experiment runs under some consideration of time and cost, and the experimenter reports that it terminates after a predetermined time or number of failures. This is called a joint censoring scheme [11]. The procedure of joint progressive type-II censoring was described previously, where the sample size S 1 + S 2 is taken as S 1 from line η 1 and S 2 from line η 2 . The integers m and r = {r 1 , r 2 , . . . , r m } are determined to satisfy the form S 1 + S 2 + m i=1 r i . The element r 1 is removed immediately from the experiment. We observe the first failure unit, say T 1 and has line W 1 from line η 1 or η 2 , say (t 1 , ω 1 ). Also, the number r 2 is removed from the test after we examine the second failure unit, say T 2 and has line W 2 , say (t 2 , ω 2 ). The experiment continues until (t m , ω m ) is observed, where w i takes the value 1 or 0, depending on lines η 1 or η 2 . The result of the previous examination t = {(t 1 , ω 1 ), (t 2 , ω 2 ), . . . , (t m , ω m )} is called the joint progressive type-II censoring procedure. The concept of a balanced joint progressive type-II censoring scheme was considered by [12] for analytically more straightforward estimators than the other type of progressive censoring procedure. Several authors have discussed statistical inference using different distributions, such as two exponential distributions [12]. The procedure of lifetime using Weibull distributions was investigated [13]. The interpretation of the balanced joint progressive type-II censoring procedure starts with samples of size S 1 + S 2 , taken from production lines η 1 and η 2 , respectively. Integers m and the integers r = {r 1 , r 2 , . . . , r m } are determined to satisfy m + m−1 i=1 r i < min(S 1 , S 2 ). The failure times and types are observed, say (t i , ω i ), i = 1, 2, . . ., m. Fig. 1 shows the main idea of a joint progressive type-II censoring scheme. This study discusses the properties of Chen lifetime estimation procedures under a joint progressive type-II censoring scheme. The Chen lifetime distribution with two parameters was introduced by [14]. This study's objective is to build a balanced joint progressive type-II censoring procedure for the Chen lifetime distribution and parameter estimation with the maximum likelihood estimator (MLE) and Bayes methods. The developed methods are also used to measure the same Chen lifetime products' relative merits under the same conditions. Estimators are evaluated through numerical data analysis and assessed through a simulation study. The remainder of this article is organized as follows. The main principle and model formulation are given in Section 2. Point MLE and interval estimators are introduced in Section 3. Section 4 discusses Bayes point and credible intervals. Estimators under numerical examples and simulation studies are discussed in Section 5. We summarize some comments which are extracted from numerical methods in Section 6.  Fig. 1 shows the scheme of joint balanced progressive type-II censoring. The observed data t = {(t 1 , ω 1 ), (t 2 , ω 2 ), . . . , (t m , ω m )} are called balanced joint progressive type-II censoring samples. Under consideration that S 1 comes from the line η 1 , and it has independent and identically distribution of lifetimes {X 1 , X 2 , . . . , X s 1 } and S 2 comes from the line η 2 , and ithas independent and identically distribution of lifetimes {X * 1 , X * 2 , . . . , X * s 2 }. These samples distributed with populations have probability density (PDFs) and cumulative distribution (CDFs) functions are given, respectively, by the functions f j (.) and F j (.), j = 1, 2. Then the balanced joint progressive type- where ω i takes the value 1 or 0, depends on line η 1 or where and R j (.) and h j (.) are reliability and hazard rate functions, respectively. Under the described model, the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the tested unit and chosen from two lines η 1 and η 2 have Chen lifetime distributions with PDFs given by Reliability and hazard rate functions, respectively, are given by and where α j and λ j are the respective shape and scale parameters of the Chen distribution. Hence, a bathtub-shaped failure rate is noticed when α j ≥1, and an exponential form can be obtained when α j = 1 [15]. Fig. 2d plots the properties of the Chen distribution. It is clearly seen that h (t) provides a bathtub-shaped curve when α = 1.

Maximum Likelihood Estimation
The joint likelihood function in Eq. (1) without a normalized constant under a Chen lifetime distribution is defined as After taking the logarithms of both sides, the joint likelihood function in Eq. (7) becomes which is used to represent the point and interval estimators of underlying parameters.

Asymptotic Confidence Interval
To obtain interval estimates of unknown parameters requires the computation of the Fisher information matrix, which is defined by the negative expectation of the partial second derivative of the log-likelihood rule using (8), where θ = (α 1 , α 2 , λ 1 , λ 2 ). In practice, the Fisher information matrix with a large sample can be approximated using the approximate information matrix, Therefore, under the rule of asymptotic normality distribution of computingθ = (α 1 ,α 2 ,λ 1 ,λ 2 ) with mean (α 1 , α 2 , λ 1 , λ 2 ) and variance covariance matrix 0 . The approximate confidence intervals for model parameters are defined aŝ where the diagonal of the approximate variance-covariance matrix 0 represents the values e 11 , e 22 , e 33 , and e 44 , and Z γ has a standard normal distribution with right-tail probability γ . The other variances are obtained using the partial derivative of the log-likelihood rule in Eq. (8), and

Bayes with MCMC Methods
We need to use Bayes approaches with the MCMC method because of the dimensionality of the model. Bayes estimation requires prior information about the model parameters, which are considered in this study to be independent gamma priors. Then, the available prior information is modeled as where θ = (α 1 , α 2 , λ 1 , λ 2 ). The joint distribution of prior densities is formed by Following this, the information about the model parameters is obtained from the prior information and the data, which provides the posterior distribution as where the denominator of the fraction can be removed since it contains no information about θ. The proportional form from posterior distribution (26) with prior distribution (25) and likelihood rule (7) is defined as The Bayes estimators are computed with respect to the loss rule; then the Bayes method of any function π (α 1 , α 2 , λ 1 , λ 2 ) under the rule of the squared-error loss (SEL) function is presented bŷ The integrals in Eqs. (26) and (28) generally cannot be obtained in explicit form, but can be solved by approximation, such as numerical integration or Lindley approximation. One of the most frequently applied methods is the MCMC method, which is used to compute point and interval estimates as follows. The full conditional distributions can be described as and Then the full conditional distributions are reduced to gamma distributions represented by Eqs. (31) and (32), and two distributions similar to normal distributions, shown as Eqs. (29) and (30). The MCMC methods have the forms of Gibbs algorithms, and the more general Metropolis-Hastings (MH) under Gibbs algorithms [17]. The following algorithm describes MCMC methods.
Step 3: The values α (ρ) j , j = 1,2 are generated from conditional distributions presented by Eqs. (29) and (30) with the MH algorithm using normal proposal distributions with mean α (ρ−1) j and variance obtained from approximate information matrix, respectively.
Step 5: Steps (2) to (4) are repeated S times. Step 6: If we need to the number of iterations to reach convergence in the equilibrium, which called burn-in, say S * ; hence, the Bayes estimators of model parameters are represented bŷ Step 7: The 100 (1 − 2γ ) % credible intervals can be obtained from the empirical distribution of θ i after putting the values in ascending order; hence, a credible interval is formed by where θ = (α 1 , α 2 , λ 1 , λ 2 ).

Simulation Studies
Two estimation methods, classical ML and Bayes estimation under Chen lifetime distribution, are discussed and developed in this study. We compare and assess these methods under the MCMC algorithms. We report the results with various sample sizes (S 1 , S 2 ), several sample sizes of failure units m, and censoring procedures r. We fix parameters at (α 1 , λ 1 ) = (0.5, 0.5) and (α 1 , λ 1 ) = (0.7, 0.4). The validity of numerical results is determined by the mean value (MV) and mean squared-error (MSE) for point estimators. The probability coverage (PC) and average interval length (AL) are used to measure interval estimators. The results are summarized in Tabs. 1 and 2 for two sets of prior information (non-informative prior 0 and informative prior 1). The simulation study used 1000 balanced progressive type-II samples. For Bayes results, the producer was considered under the rule of the squared-error loss function and 11000 iterations of MCMC, with the first 1000 iterations as burn-in. The results are reported in Tabs. 1 and 2.   Under consideration two sample of size (S 1 , S 2 ) = (40, 40), censoring scheme r = 9, 0 (28) , with the number of failures m = 30. Then the sample can be generated with sample size S 1 = 30 from a Chen distribution with parameters (1.5, 1.1) and with size S 2 from a Chen distribution with parameters (1.8, 0.9) using the algorithms [18]. The two progressive type-II samples are used to generate balanced joint progressive type-II samples with respect to r = {9, 0 (28) } and m = 30. The joint sample and its type are reported in Tab

Concluding Remarks
Products from different production lines were investigated using a joint censoring procedure under the same conditions. The balanced joint censoring procedure has been shown considerable attention over the last few years. In this study, we discussed products that follow a Chen lifetime distribution. We discussed the ML and Bayes estimates to estimate the underlying parameters of two Chen lifetime distributions. Numerical results were obtained to compare the theoretical performance results. Some points are observed from numerical results, which are summarized as follows.
From the results in Tabs. 1 and 2, show that the balanced joint progressive type-II censoring procedure provides better excellent results for products have Chen lifetime distribution.
Estimation results under the Bayes method and informative prior distribution provide better estimation than ML and non-informative prior methods according to the MSE.
For non-informative priors, there are no significant differences between MLEs and Bayes estimates.
The effective sample size m can be increased by reducing the MSEs and interval lengths.