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

Department of Mathematics & Statistics, Faculty of Science, Taif University, Taif, 21944, Saudi Arabia
*Corresponding Author: Hassan M. Aljohani. Email: hmjohani@tu.edu.sa
Received: 08 August 2020; Accepted: 29 October 2020

Abstract: 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 complicated 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.

Keywords: Chen distributions; progressive type-II censoring; maximum likelihood; mean posterior; Bayesian estimation; MCMC

1  Introduction

Figure 1: Example of the structure of joint progressive type-II censoring procedures

2  Model Formulation

Assume two production lines, and a random sample of size S1 +S2, where S1 comes from line and S2 from line . The integers m and are determined to satisfy . Suppose t1 is observed from some units that are taken from line then, r1 survival component is removed from S1 and r1 +1 survival component is removed from S2 when the second failure t2 is observed if t2 is chosen from the line In that case, r2 +1 survival component is removed from S1r1 −1, and r2 survival component is removed from the sample S2r2 −1. The test continues in this manner until the mth failure tm is observed. If the final failure is from line , then the survival components are removed from , and are removed from . If the final failure belongs to line , then the survival units are removed from , and are removed from . Fig. 1 shows the scheme of joint balanced progressive type-II censoring. The observed data are called balanced joint progressive type-II censoring samples. Under consideration that S1 comes from the line , and it has independent and identically distribution of lifetimes and S2 comes from the line , and ithas independent and identically distribution of lifetimes . These samples distributed with populations have probability density (PDFs) and cumulative distribution (CDFs) functions are given, respectively, by the functions fj(.) and , j = 1, 2. Then the balanced joint progressive type-II sample is taken from , where m = m1 +m2, m1 is the number of failed units from line , and m2 is the number of failed units from line . The observed balanced joint progressive type-II censoring sample is where takes the value 1 or 0, depends on line or , and .

The joint likelihood rule under two progressive type-II censoring samples is

where

and Rj(.) and hj(.) 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 and have Chen lifetime distributions with PDFs given by

Reliability and hazard rate functions, respectively, are given by

and

where and are the respective shape and scale parameters of the Chen distribution. Hence, a bathtub-shaped failure rate is noticed when 1, and an exponential form can be obtained when [15]. Fig. 2d plots the properties of the Chen distribution. It is clearly seen that provides a bathtub-shaped curve when .

Figure 2: Examples of the scaled Chen distribution for different values of with : (a) Chen distribution; (b) Cumulative distribution; (c) Reliability function; and (d) Hazard rate function

3  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.

3.1 MLEs

The likelihood rule is obtained from Eq. (8) by taking partial derivatives with respect to the parameter vectors and equating to zero.

The equation is reduced to

The equation is reduced to

The equation is reduced to

The equation is reduced to

After replacing in (9)(11) and in (10)(12), we obtain

and

Nonlinear Eqs. (13) and (14) with only one parameter can be solved using any iteration method such as Newton-Raphson or fixed point iteration. The parameter estimates and are obtained, and parameter estimates and are obtained from Eqs. (9) and (10) after replacing and by and . If m1 = 0 or m2 = 0, then the parameter values and or and cannot be obtained [16].

3.2 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 . 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 computing with mean and variance covariance matrix . The approximate confidence intervals for model parameters are defined as

where the diagonal of the approximate variance-covariance matrix represents the values e11, e22, e33, and e44, and has a standard normal distribution with right-tail probability . The other variances are obtained using the partial derivative of the log-likelihood rule in Eq. (8),

and

4  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 . 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 . 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 under the rule of the squared-error loss (SEL) function is presented by

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 2: The values , j = 1, 2 are generated from conditional distributions presented by Eqs. (31) and (32), respectively.

Step 3: The values , j = 1, 2 are generated from conditional distributions presented by Eqs. (29) and (30) with the MH algorithm using normal proposal distributions with mean and variance obtained from approximate information matrix, respectively.

Step 4: The vector is recorded; hence, .

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 by

with posterior variance of ,

Step 7: The credible intervals can be obtained from the empirical distribution of after putting the values in ascending order; hence, a credible interval is formed by

where .

5  Numerical Computation

5.1 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 (S1, S2), several sample sizes of failure units m, and censoring procedures r. We fix parameters at and . 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.

Table 1: MVs and MSEs of estimators of Chen distributions with

Table 2: Two ALs (PCs) of Chen distributions with

5.2 Data Analysis

Let Chen distribution with parameter values and and the prior distributions with parameters (a1, b1) = (4, 2), (a3, b3) = (2.0, 1.5) and (a4, b4) = (2, 2.5) are used to apply Bayes approaches.

Under consideration two sample of size (S1, S2) = (40, 40), censoring scheme with the number of failures m = 30. Then the sample can be generated with sample size S1 = 30 from a Chen distribution with parameters (1.5, 1.1) and with size S2 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 and m = 30. The joint sample and its type are reported in Tab. 3. The results of point estimation and interval MLEs are reported in Tab. 4. We plot the monitoring of the MCMC and the corresponding histogram in Figs. 310, which show the quality of the empirical posterior distribution generated by MCMC methods.

Table 3: Balanced joint progressive type-II censoring data

Table 4: Point and 95% confidence and credible intervals (ACIs and CIs)

Figure 3: Recording of parameter generated by the MCMC algorithm

Figure 4: Summary of the analysis for generated by the MCMC algorithm

Figure 5: Recording of parameter generated by the MCMC algorithm

Figure 6: Summary of the analysis for generated by the MCMC algorithm

Figure 7: Recording of parameter generated by the MCMC algorithm

Figure 8: Summary of the analysis for generated by the MCMC algorithm

Figure 9: Recording of parameter generated by the MCMC algorithm

Figure 10: Summary of the analysis for generated by the MCMC algorithm

6  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.

Acknowledgement: The researcher would like to thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript. This study was funded by Taif University Researchers Supporting Project number (TURSP-2020/279), Taif University, Taif, Saudi Arabia.

Funding Statement: Taif University.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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