Bayesian Estimation Based on Rayleigh Progressive Type II Censored Data with Binomial Removals

This study considers the estimation problem for the parameter and reliability function of Rayleigh distribution under progressive type II censoring with random removals, where the number of units removed at each failure time has a binomial distribution. We use the maximum likelihood and Bayesian procedures to obtain the estimators of parameter and reliability function of Rayleigh distribution.We also construct the confidence intervals for the parameter of Rayleigh distribution.Monte Carlo simulationmethod is used to generate a progressive type II censored data with binomial removals from Rayleigh distribution, and then these data are used to compute the point and interval estimations of the parameter and compare both the methods used with different random schemes.


Introduction
The Rayleigh distribution provides a population model which is useful in several areas of statistics.In the literature, many researcher studied properties of the Rayleigh distribution, particularly in life testing and reliability.Life testing experiments often deal with censored sample in order to estimate the parameters involved in the life distribution.The cumulative distribution function (CDF), probability density function (PDF), and the reliability function of the Rayleigh distribution with parameter  > 0 are respectively.Inferences for  in the Rayleigh distribution have been discussed by several authors.Harter and Moore [1] derived an explicit form for the maximum likelihood estimator (MLE) of  based on type II censored data.Moreover, Bayesian estimation and prediction problems for the  based on doubly censored sample have been considered by Fernández [2] and Raqab and Madi [3].Wu et al. [4] have derived the Bayesian estimator and prediction intervals based on progressively type II censored samples.A recent account on progressive censoring schemes can be obtained in the monograph by Balakrishnan and Aggarwala [5] or in the excellent review article by Balakrishnan [6].Suppose that  units are placed on a life test and the experimenter decides beforehand quantity , the number of units, to be failed.Now at the time of the first failure,  1 of the remaining  − 1 surviving units are randomly removed from the experiment.Continuing, at the time of the second failure,  2 of the remaining  −  1 − 2 units are randomly drawn from the experiment.Finally, at the time of the th failure, all the remaining

Process
The number in life testing Failures Binomial removals Remains −1 surviving units are removed from the experiment.Note that, in this scheme,  1 ,  2 , . . .,   are all prefixed.However, in some practical situations, these numbers may occur at random.For example, in some reliability experiments, an experimenter may decide that it is inappropriate or too dangerous to carry on the testing on some of the tested units even though these units have not failed.In such cases, the pattern of removal at each failure is random [7,8].This leads to progressive censoring with random removals (illustrated by Table 1).There have been several references about the statistical inference on lifetime distributions under progressive censoring with random removals.Wu et al. [9] Wu and Chang [10] considered the estimation problem for the Burr type-XII distribution and Pareto distribution based on progressively censored samples with random removals, where the number of units removed at each failure time has a discrete uniform removal pattern and binomial distribution.

Maximum Likelihood Estimation
Let  1 <  2 < ⋅ ⋅ ⋅ <   be the ordered failure times out of  randomly selected times, where  is predetermined before the test.At the th failure,   items are removed from the test.For progressive censoring with predetermined number of removals  = ( 1 =  1 , . . .,  −1 =  1 ), the likelihood function can be defined as the following form: where 4) is derived conditional on   .Each   can be of any integer value between 0 and −−∑ −1 =1 (  ).It is different from progressive censoring with fixed removal that   is a random number and is assumed to follow a binomial distribution with parameter .It means that each unit leaves with equal probability  and the probability of   units leaving after the th failure occurs is where 0 ≤   ≤  −  − ∑ −1 =1   ( = 1, . . .,  − 1).Furthermore, we assume that   is independent of   for all .The joint likelihood function of  = ( 1 ,  2 , . . .,   ) and  = ( 1 ,  2 , . . .,   ) can be found as where (, ) is the joint probability distribution of  = ( 1 ,  2 , . . .,   ) and in particular Therefore Substituting ( 1) and ( 2) into ( 6), the likelihood function takes the following form: Thus, the log-likelihood function takes the following form: The first partial derivatives of log-likelihood function  with respect to  and  are Solving these equations in  and , we get the MLE of  and  as in the following forms: It is clear that the MLE of reliability function, ( 0 ), with fixed  0 > 0, can be obtained by
Confidence interval for  2 can be obtained through the pivotal quantity 2σ 2 / 2 ∼  2 (2) and can be the percentile of Chi-square distribution with 2 degrees of freedom and right-tail probability .Then since  2 is increasing function of .Then confidence interval for  is ) . (16)

Bayesian Estimation
In this section, we use the Bayes procedure to derive the point estimates of the parameters  and  and reliability function ( 0 ) based on progressively type II censoring data with binomial removals.For this purpose we assume the parameters  and  behave as independent random variables.In this paper, for parameter  we consider prior distribution of the form where  > 0 and  > 0. This density is known as the square-root inverted-gamma distribution (see [12, page 119]) and, independently from parameter ,  has a beta prior distribution with parameters  and  and PDF Based on the prior  1 () and  2 (), the joint prior PDF of (, ) is It follows, from (9) and (20), that the joint posterior distribution of (, ) is given by where Therefore, the marginal posterior PDFs of  and  are given, respectively, by Substituting  2 = − 2 0 /2 log  into (22), we obtain the posterior probability density function of  = ( 0 | ) as for 0 <  < 1.

Bayesian Estimation under Squared Error Loss Function.
Under squared error loss, the Bayes estimator of  is the mean of posterior density (22), which is given by And for parameter , the Bayes estimator of  is Finally, the Bayes estimators of ( 0 ) are found to be

Bayesian Estimation under Entropy Loss Function.
In many practical situations, it appears to be more realistic to express the loss in terms of the ratio θ/,  = (, ).In this case, Calabria and Pulcini [13] point out that a useful asymmetric loss function is the entropy loss function: where whose minimum occurs at θ = .The latter has been used in Dey et al., [14] and Dey and Liu [15], in the original form having  = 1: Therefore, the Bayesian estimators of parameters  = (, ) under the entropy loss function with  = 1, denoted by θ , is given by the following equation: Therefore we have

Bayesian Estimation under Precautionary Loss Function.
Precautionary loss function which is asymmetric, was introduced by Norström [16]: The Bayes estimator under the latter loss function is denoted by θ and may be obtained by solving the following equation: The Bayesian estimators of the parameters ,  and () under asymmetric precautionary loss function are given, respectively, by (39)

Numerical Study
In our study, we firstly generate the numbers of progressive censoring with binomial removals   ( = 1, 2, . . ., ), and then we get the progressive censoring with binomial removals samples from Rayleigh distribution by the Monte-Carlo method.The steps are as follows.
Step 1. Generate a group values according to the relevant value of .
Step 3. Set for  = 1, 2, . . ., .This is the required progressively type II censored sample with binomial removals from the standard exponential distribution.Step 5. We compute the MLE of parameters  and  by (12), for reliability function by (13) and 95% confidence intervals by (16).
Step 7. We repeat the above steps 2000 times.We then obtain the means and the MSEs (mean-squared error) for different censoring sizes  and censoring schemes where and φ is the estimator of .
In all above cases the prior parameters chosen as ( = 2,  = 1) and ( = 3,  = 2) which yield the generated value of  = 1 as the true value.The true values of ( 0 ) in  0 = 0.5 is obtained (0.5)= 0.8824969.The results are summarized in Tables 2 and 3.

Conclusions
We develop some results on Rayleigh distribution when progressive type II censoring with binomial removals is performed.Based on the results shown in

Table 1 :
A schematic representation of the progressive type II censoring with binomial removals.

Table 3 :
Averaged values of MSEs for estimates of the reliability function.

Table 2
, one can conclude that Bayesian estimators under precautionary loss function have the smallest biased and MSE as compared with the other estimators.The MSE associated with the estimators (Bayesian estimators under different loss functions and maximum likelihood estimators) of the parameter  decrease with increasing the sample size .According to Table 3 maximum likelihood estimators of reliability function of Rayleigh distribution have smallest biased and MSE as compared with the other estimators.It is immediate to note that MSEs of reliability function estimators decrease as ,  increases.