Bayesian Estimation and Prediction Based on Progressively First Failure Censored Scheme from a Mixture of Weibull and Lomax Distributions

This paper develops Bayesian estimation and prediction, for a mixture of Weibull and Lomax distributions, in the context of the new life test plan called progressive first failure censored samples. Maximum likelihood estimation and Bayes estimation, under informative and non-informative priors, are obtained using Markov Chain Monte Carlo methods, based on the symmetric square error Loss function and the asymmetric linear exponential (LINEX) and general entropy loss functions. The maximum likelihood estimates and the different Bayes estimates are compared via a Monte Carlo simulation study. Finally, Bayesian prediction intervals for future observations are obtained using a numerical example.


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The corresponding cumulative distribution function (cdf) and reliability function, respectively are given by Common censoring schemes of type I and type II censoring do not allow units to be removed from the test at any other point than the final termination point. Therefore, the focus in the last few years has been on progressive censoring due to its flexibility that allows the experimenter to remove active units during the experiment. Progressive censoring has been studied by many authors. Some of the early work on progressive censoring are Cohen (1963), Mann (1971), Viveros and Balakrishnan(1994), Balasooriya et al. (2000), Balakrishnan and Aggarwala (2000) and Balakrishnan (2007) have presented an elaborate overview of various developments in progressive censoring data.
The experimental time is usually an important concern for the life test designers. Although conventional censoring scheme can shorten the duration of a life test, the experimental time is still too long that cannot be waited for when the units are highly reliable. Johnson (1964) introduced the first failure censoring by grouping the test units into several sets. The experimenter runs all test units simultaneously until the first failure in one of the sets. Some references that discussed this type of first failure are Balasooriya (1995), Wu et al. (2003) and Wu and Yu (2005). Wu  The objective of this work is to apply the Bayesian procedure to estimate the parameters and obtain two sample prediction bounds for future observations from the proposed model, based on progressive first failure censoring scheme. The rest of this paper is organized as follows: In Section 2, the progressive first-failure censoring scheme is described. In Section 3, we obtain maximum likelihood estimators of the parameters. Different loss functions are presented in Section 4. The Bayesian estimation is discussed in Section 5, using Markov Chain Monte Carlo technique. In Section 6, Monte Carlo simulation study is conducted to compare the performance of different estimation methods. Bayesian prediction with numerical data are presented in Section 7. Finally, we conclude the paper in Section 8.

A Progressive First-Failure Censoring Scheme
In a life-testing experiment, suppose sample of independent groups with items in each group are put in a life test. After units have failed, each item can be attributed to the appropriate subpopulation. Thus if the are failed during the interval (0, ( ) ): 1 from the first subpopulation and 2 from the second subpopulation. When the first failure is observed, 1 groups and the group with observed failure are randomly removed. At the second observed failure, 2 groups and the group with observed failure are randomly removed. This experiment terminates at the time when the ℎ failure is observed and the remaining groups and the group with observed failure are all removed. Here 1, , , < 2, , , < ⋯ < , , , , are known as progressive first failure censored orders statistics with the progressive censoring scheme = ( 1 , 2 , … , ); ( ≤ ), Let denote the failure of the ℎ unit that belongs to the ℎ subpopulation and ≤ ( ) ; = 1,2, … , ; = 1 + 2 . where ( ) denotes the failure time of the ℎ unit. For a two component mixture model, the likelihood function, is defined as ), 1 = , 2 = 1 − . Note that if = 1 then, sampling scheme reduces to the progressively type II censoring, a first failure censored scheme when = (0, … , 0), a usual type II censored scheme when = 1 and = (0, … ,0), and complete sample case if = 1 and = (0, … ,0), with = . It should be noted that 1, , , , 2, , , , … , , , , can be viewed as a progressive type II censored sample from a population with distribution function 1 − (1 − ( )) . For this reason, results extended to progressive first-failure censored scheme easily. The progressive first-failure censored plan has advantages in terms of reducing the test time, in which more items are used but only of × items are failures.

Loss Function
In decision theory, the loss criterion is specified in order to obtain the best estimator. Three loss functions are proposed, symmetric (square error) loss function and asymmetric (LINEX and general entropy) loss functions, as follows: • Squared error loss function: A simple, and very common loss function is defined by 1 (̂, ) = (̂− ) 2 ; c is a constant which is symmetrical in nature and gives equal weight to overestimation as well as under estimation. However, in real applications, estimation of reliability and failure rate function, an overestimate is more serious than the underestimates. The use of asymmetric loss function might be inappropriate as has been recognized by Basu and Ebrahimi (1991).
• Linear exponential loss function (LINEX): One of the most commonly used asymmetric loss functions, introduced by Varian (1975) under the assumption that the minimal loss occurs at ̂= , it can be expressed as where determines the shape of the loss function. If > 0 means overestimation and underestimation if < 0, but in a situation where ≅ 0, the LINEX loss is almost symmetric and approaches square error loss function.
Under the above loss function, the Bayes estimator ̂ of can be obtained as provided that the expected value with respect to the posterior function of , ( − | ) exists and is finite.
• General entropy loss function: Another commonly asymmetric loss function is the modified LINEX loss function called a general entropy loss function proposed by Calabria and Pulcini (1996) which has a minimum at ̂= . Also, the loss function used by several authors, in the original form having the shape parameter ℎ = 1, for ℎ > 0, a positive error has a more effect than a negative error. In this case, the Bayes estimate of is given by provided that the expected value with respect to the posterior function of , ( −ℎ | ) exists and is finite.

Bayesian Estimation
In this section, we derive Bayes estimators of the parameters 1 , 2 and of the considered model based first failure progressively censored sample. Assuming the following independent prior distributions for the parameters 1~( 1 , 1 ), 2~( 2 , 2 ), and ~( , ) for the mixing parameter . The joint prior distribution of 1 , 2 and is The joint density function of 1 , 2 , and the sample can be written as follows Based on Equation (9) , the joint posterior density function of 1 , 2 and , is given by Thus, under the squared error, LINEX and general entropy loss functions, the Bayes estimators of any function of ( , 1 , 2 ), say ( , 1 , 2 ), is The ratio of the integrals in Equations (11), (12) and (13) cannot be obtained in a closed form. Therefore, the Markov Chain Monte Carlo technique will be used to approximate the integrals.

Markov Chain Monte Carlo method (MCMC)
In this subsection, we apply the importance sampling technique to obtain the approximate Bayes estimates. This technique needs no calculation of the normalizing constant. The joint posterior density function of 1 , 2 and given data can be written in the form where 1 , 2 are defined as above, and According to the importance sampling technique, the approximate Bayes estimators, based on the three loss functions, can be computed by the following algorithm Step1: ~( 1 + , 2 + ) Step2: Repeat this procedure to obtain important sample {( , 1 , 2 ), = 1, … , } Step 3: Calculate Bayes estimator of ( , 1 , 2 ), under squared error, LINEX and general entropy loss functions, respectively by

Comparison Study
In this section, we present the numerical results of a simulation study to compare the performance of the various estimates for different combinations of ( , , ). We consider different sampling schemes = ( 1 , … , ) as follow To carry out this comparative study, we follow the following steps: • In this study the following parameters values were used (  • Appling the algorithms of Balakrishnan and Sandhu (1995) to generate a progressive first-failure censored sample from mixture Weibull and Lomax distributions for different values of ( , , , ).
• Based on the progressive first-failure censored data, maximum likelihood estimates and Bayes estimates of the parameters are calculated according to Section 3 and Section 5, respectively.
• The above steps are repeated 1000 times, and computation off the average of the estimates and mean square error for different values ( , , , ) are presented in Tables (2-8). The computations are done using Mathematica 10.0.
Table (2)(3)(4)(5)(6)(7)(8) indicates that the Bayes estimates perform better under informative prior than non-informative prior for all different loss functions. In most cases, notice that the performance of the Bayes estimates under informative prior are smaller than the maximum likelihood estimates in terms of MSE. Also, the Bayesian estimates under general entropy loss function in case of the value (ℎ = −1) are almost the same as the estimates under squared error loss function. The mean square error of all estimates decreases when the sample size and effective sample size increase. Also, when the value of the group size increases, the mean square error decreases.

Bayesian Prediction For Future Observations
In this section, the Bayesian two sample prediction of a future order statistics is considered based on the observed progressive first failure censored data . Based on a random sample of size drawn from a population with pdf (1), a future unobservable independent random sample of size from the same population is under consideration. Let represents the ℎ ordered statistic in the future sample, 1 ≤ ≤ . The ℎ order statistic in a sample of size represents the life length of a ( − + 1) out of system. The distribution function of the ordered future sample is given, [See Arnold et al.(1992) and Jaheen (2003)], by where ( | , 1 , 2 ) is the cumulative distribution of the ℎ component in a future sample as given by (17). This cannot be evaluated analytically. Thus, the MCMC sampling procedure described in Subsection 5.1 is applied. These equations cannot be solved analytically, using Mathematica software program.

Numerical Example
This section presents a numerical example to illustrate the methodology for the proposed estimates based on real data. The data set is an uncensored data set consisting of 66 observations on breaking stress of carbon fibers (in Gba Based on these data, we compute predictive interval of future sample. According the algorithm in Subsection 5.1, we generate 1000 MCMC samples. The 90% and 95% predictive interval for the future observation are given by solving Equation (21) numerically. The results are presented in Table (9) and Table (10)

Conclusion
Based on progressively first-failure censored scheme, in this paper, we have addressed the estimation and prediction problems of the mixture of Weibull and Lomax distributions. The Bayes estimates cannot be obtained in explicit form so importance samples procedure is used to draw MCMC samples. Also, the same MCMC method is used for computing two sample predictive intervals. An example using real data set was used for illustration.