The Comparison Between Standard Bayes Estimators of the Reliability Function of Exponential Distribution

In this paper, a Monte Carlo Simulation technique is used to compare the performance of the standard Bayes estimators of the reliability function of the one parameter exponential distribution .Three types of loss functions are adopted, namely, squared error loss function (SELF) ,Precautionary error loss function (PELF) and linear exponential error loss function (LINEX) with informative and noninformative prior .The criterion integrated mean square error (IMSE) is employed to assess the performance of such estimators .


Introduction
The reliability theory is associated with random occurrence of undesirable events or failure during the life of a physical or biological system [1].Reliability is a substantial feature of a system.Basic concepts related with reliability has been recognized for a number of years, however, it has got greatest importance during the past decennium as a consequence of the use of highly complex systems.In reliability theory, the exponential distribution has a distinctive role in life testing experiments.Historically, it was the first life time model for which statistical procedures were widely developed.Many researchers gave numerous results and generalized the exponential distribution as a life time distribution, particularly, in the field of industrial life testing.The exponential distribution is desirable because of its simplicity and its own features such as lacks memory and self-producing property.The probability density, cumulative distribution and reliability functions of one parameter exponential distribution are respectively defined as [2]:  , ɵ ɵ ɵ , , ɵ 0 1 The cumulative distribution function is given by F t pr T t 1 e ɵ 2 R t 1 F t e ɵ 3 The one parameter exponential distribution is a member of exponential class of probability density functions which has the general form [3] f t, ɵ exp p ɵ k t s t q ɵ 4 then the exponential distribution the p.d.f.can be written as f t, ɵ exp ɵ ɵ Hence T ∑   ∑  is the complete sufficient statistic for ɵ and it can be easily shown that T follows Gamma distribution with parameters n and ɵ .

Standard Bayes Estimators
The researchers employed three types of loss functions, namely, the squared error loss function (SELF), precautionary error loss function (PELF) and linear exponential error loss function(LINEX) .The Bayes estimator of the parameter ɵ is the value of ɵ that minimize the risk function R(ɵ , ɵ where [ ThePrecautionary error loss function is defined as [5]: If (PELF)is adopted, it can be in the same manner show that the Bayes estimator of ɵ is Varian (1975) developed the following a symmetric linear exponential (LINEX) loss function  ∆  ∆ ∆ 1 10 Where ∆ ɵ ɵ And when the ((LINEX) is adopted, similarity the Bayes estimator of ɵ is ɵ   ɵ ℎ ɵ‫|‬  11 ɵ

Posterior Density Based on Jeffrey's Prior Information
Let us assume that ɵ has non informative prior density.Jeffrey's (1961) developed a general rule for obtaining the prior distribution of ɵ [6].He established that the single unknown parameter ɵ which is regarded as a random variable follows such a distribution that is proportional to the square root of the fisher information on ɵ, that is [ The posterior density in Equation ( 14) is defined identified as a density of the Gamma distribution, that is ɵ‫|‬ ,  , … ,  ~ Gamma (n, ) with E(ɵ) = and var(ɵ) = that is ɵ~Gamma (n , )

Posterior Density Based on Gamma Prior Distribution
Assuming that ɵ has informative prior as Gamma distribution which takes the following form: The simulation study was conducted in order to compare the performance of the Bayesian estimators of the reliability function R(t)of one parameter exponential distribution.
The integrated mean squared error (IMSE) as a criterion of comparison where Where  is the random limitsof  , using t=(0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1)L is the number of replications which we assumed that L=1000 in our study,  ( ) is the estimator of R(t) at the  replication.The Bayesian estimators of R(t) are derived with respect to three loss function which are the square error loss function (SELF),precautionary error loss function (PSELF) and linear exponential error loss function(LINEX) , moreover, the informative and non-informative prior were postulated .The sample sizes n=10,50, 100 and 200 were chosen to represent small, moderate, large and very large sample sizes from the one parameter exponential distribution .The postulated values of the unique parameter ɵ were ɵ=0.5,1.5 and the values of the parameters for Gamma prior were α=0.3,1 and β=1.2,3 .

Table 1 .
(IMSE's) values of the reliability function estimators by using Jeffrey's prior information at ɵ=0.5

Table 2 .
(IMSE) values of the reliability function estimators by using Jeffrey's prior

Table 3 .
(IMSE's) of the reliability function estimators by using Gamma prior information at ɵ=0.5

Table 4 .
(IMSE's) of the reliability function estimators by using Gamma prior information at ɵ=1.5.

Table 1 .
and Table 2.: The Bayes estimator under  Precautionary error loss function with Jeffrey's prior is the best comparing to other estimators for all sample sizes. From Table 3.: The Bayes estimator under precautionary error loss function with Gamma prior (α=0.3,β=3) is the best comparing to other estimators for all sample sizes. From Table 4.: for (n=10) the performance of Bayes estimator under squared error loss function with Gamma prior (α=1, β=1.2) is the best, and for (n=50,100) the performance of Bayes estimator under precautionary error loss function with Gamma prior (α=0.3,β=3) is the best and for (n=200) the performance of Bayes estimator under precautionary error loss function with Gamma prior (α=1, β=3) is the best. That is the performance of Bayes estimator under precautionary error loss function is superior to the performance of other estimators in almost cases that are studied in this paper, where the integrated mean squared error (IMSE) is employed as a criterion to assess the performance of such estimators.