Comparison of Bayes ' Estimators for the Exponential Reliability Function Under Different Prior Functions Jinan

In this study, we derived the estimation for Reliability of the Exponential distribution based on the Bayesian approach. In the Bayesian approach, the parameter of the Exponential distribution is assumed to be random variable .We derived posterior distribution the parameter of the Exponential distribution under four types priors distributions for the scale parameter of the Exponential distribution is: Inverse Chi-square distribution, Inverted Gamma distribution, improper distribution, Non-informative distribution. And the estimators for Reliability is obtained using the two proposed loss function in this study which is based on the natural logarithm for Reliability function .We used simulation technique, to compare the resultant estimators in terms of their mean squared errors (MSE).Several cases assumed for the parameter of the exponential distribution for data generating of different samples sizes (small, medium, and large). The results were obtained by using simulation technique, Programs written using MATLAB-R2008a program were used. In general, we obtained a good estimations of reliability of the Exponential distribution under the second proposed loss function according to the smallest values of mean squared errors (MSE) for all samples sizes (n) comparative to the estimated values for MSE under the first proposed loss function.


Introduction
The exponential distribution is one of the most important distributions in life-testing and reliability studies.Inference procedures for the exponential distribution and applications in the context of life-testing and reliability have been discussed by many authors.We mention some of them in a brief manner: Chiou (1993) [1] proposed two empirical Bayes shrinkage estimators for the reliability of the exponential distribution and study their properties.Under the uniform prior distribution and the inverted gamma prior distribution these estimators are developed and compared with a preliminary test estimator with a shrinkage estimator in terms of mean squared error.Baklizi (2003) [2] investigated the advantages of incorporating prior information in the reliability function through the shrinkage estimators.His work is an effort to coin unified shrinkage estimators of reliability function for five lifetime distributions commonly used to model lifetime data by biologists, physicists, engineers and statisticians for living and non-living entities.Sarhan (2003) [3] exploits past experiments to approximate a prior information (prior density) into the model, in estimating reliability function and parameter of exponential distribution, by using bayesian approach.Balakrishnan, Lin and Chan( 2005) [4] made a comparison of these two prediction intervals based on the expected width of the prediction interval, as well as by means of the probability of the width of one being smaller than the other.Friesl and Hurt (2007) [5] gave some basic ideas of both the construction and investigation of the properties of the Bayesian estimates of certain parametric functions of the exponential distribution under the model of random censorship assuming the Koziol-Green model.Various prior distributions are investigated and the corresponding estimates are derived.Liu and Ren (2013) [6] studied the empirical Bayes estimation of the parameter of the exponential distribution.In the empirical Bayes procedure, they employ the non-parameter polynomial density estimator to the estimation of the unknown marginal probability density function, instead of estimating the unknown prior probability density function of the parameter.They Empirical derived bayes estimators for the parameter of the exponential distribution under squared error and LINEX loss functions.So in this paper, we try to find best estimation for Reliability function ( R(t)) of exponential distribution which it means the probability of surviving at least till age t.According to the smallest value of Mean Square Errors (MSE) were calculated to compare bayes estimators under four types of prior distributions to get bayes estimation :Inverse Chi-square distribution , Inverted Gamma distribution, Improper distribution ,Non-informative distribution when the Bayesian estimation is based on two proposed loss functions .Several cases from exponential distribution for data generating ,for different sample sizes (small, medium, and large).The results were obtained by using simulation technique, Programs written using MATLAB-R2008a program were used.

Exponential Distribution
We consider t1, t2, …, tn is a random sample of n independent observations from an Exponential distribution having the probability density function (pdf) define as [7]: where θ > 0 is mean, standard deviation, and scale parameter of the distribution, θ is a survival parameter in the sense that if a random variable t is the duration of time that a given biological or mechanical system manages to survive, and t ~ Exp( θ ) then the expected duration of survival of the system is θ units of time .So the cumulative (distribution) function is Where R(t) is probability of surviving at least till age t .And F (t) is the cumulative distribution function.

Bayes Estimation Method
In this section, we used several methods to estimate Reliability function(


be a random sample of size n with probability density function given in equation ( 1) and likelihood function from the Exponential pdf given in (1) will be as follows [7]: Substituting the equation ( 4) and for each ) θ P( as shown in the table above in equation ( 5), we get the posterior distributions for the unknown parameter θ are derived using the following four types of priors ( for more details see Appendix-A).
The posterior distributions ( t) \ θ P( ) for the unknown parameter (θ) are derived using the following four types of priors.Prior dist n .

2-Bayes' Estimators
Bayes' estimators for Reliability function (R= R(t) ), was considered with four different priors and under two loss functions proposed: 1.The first proposed loss function

The second proposed loss function
Where R an estimator for is R , was considered with different four priors, and under two loss functions proposed.The following is the derivation of these estimators:

-The first proposed loss function
To obtain the Bayes' estimator, we minimize the posterior expected loss given by: under the squared error loss function with different four priors.
Prior distribution ) Γ(. is a gamma function.

-The second proposed loss function
To obtain the Bayes' estimator, we minimize the posterior expected loss given by:  ) for each sample size (n).with MSE for all sample size (n) and values (v, ( α , β ), (a , b), c ) respectively.Also the results of the simulation study are summarized and tabulated in tables (5 to 8) see appendix-E .In each row of tables (5 to 8) ,we have four estimated values for R(t) ^with MSE for all sample size (n) and values (v, ( α ,β ), (a, b), c) respectively .The Bayes estimators under four types of prior distribution .So our criteria is the best estimator that gives the smallest value of MSE.We list the results in the tables (1 to 8) in appendix-E.

Discussion
In general, as we see in the tables ( 1 to 8 ) by using different estimation methods, See appendix-E.We find the Mean Square Errors (MSE) was decreased when sample size increased in all cases.That means the estimation of ^ ) (R(t) get better for the large sample sizes.We obtained a good estimation according to the smallest values of MSE for all samples sizes (n) comparative to the other estimated values for MSE under the first proposed loss function.As we see in table -1, when the true value of θ ( 0.5 θ  ) and the prior distribution for θ is  Inverted Gamma distribution with 2) β 5, α (   for t=0.5.  Improper distribution with (a=9, b=3) for t=1.5, 2.5, 3.5.
And we see in table -2, when the true value of θ ( 1 θ  ) and the prior distribution for θ is  Non-informative distribution with c=1 for t=0.5, 1.5.
 Inverse Chi-square distribution with v=4 for t=2.5. ) and the prior distribution for θ is  Non-informative distribution with c=1 for t=0.5, 1.5, 2.5.
 Inverse Chi-square distribution with v=4 for t=3.5.
And we see in table -4, when the true value of θ ( 2.5 θ  ) and the prior distribution for θ is Non-informative distribution with c=1 for all.See appendix-E.We obtained a good estimation according to the smallest values of MSE for all samples sizes (n) comparative to the other estimated values for MSE under the second proposed loss function.
And we see in table -6, when the true value of θ ( 1 θ  ) and the prior distribution for θ is  Non-informative distribution with c=1 for t=0.5.
And we see in table -7, when the true value of θ ( 1.5 θ  ) and the prior distribution for θ is  Non-informative distribution with c=1 for t=0.5, 1.5.
 Inverse Chi-square distribution with v=4 for t=2. ) and the prior distribution for θ is  Non-informative distribution with c=1 for t=0.5, 1.5, 2.5.
 Inverse Chi-square distribution with v=4 for t=3.5.
See the summary of discussion for ) ) t ( R MSE( ^in tables ( 9) and (10) in Appendix-E.

Conclusion
When we compared the estimated values for Reliability( R(t)) of the Exponential distribution by using Bayes with respect to Mean Square Errors (MSE) of estimated exponential reliability function under the two proposed loss function in this study .We find that MSE is decreasing when sample size is increasing in all cases.).See tables ( 9) and (10) in Appendix-E.Also ,we obtained ).See tables (9)

4 .
The number of replication used was ( 1000 L  And we see in table -3, when the true value of θ (

a
good estimation according to the smallest values of MSE for all samples sizes (n) comparative to the other estimated values for MSE under the second proposed loss function, when the prior distribution for θ is  Non-informative distribution with c=1 for t=0.5, and inverted Gamma distribution with t= 1.5, 2.5 when the true value of θ ( 1 θ  ). Non-informative distribution with c=1 for t=0.5 and inverse Chi-square distribution with v=4 for t=2.5, and inverted Gamma distribution with So, the following results are the derivations of these estimators under the first proposed loss function with different four priors (for more details see Appendix-B).
So, the following results are the derivations of these estimators under the second proposed loss function with different four priors (for more details see Appendix-C).
^) under the squared error loss function with different four priors.
See appendix-D, for the programs algorithm.The results of the simulation study are summarized and tabulated in tables (1 to 4) see appendix-E.In each row of tables (1 to 4) ,we 6.We obtained estimators for Reliability function ( R(t) ), the estimators in the table in section (3.2.1), it means the estimators 5.

Table ( 1): MSE of estimated exponential reliability function under the first proposed loss function. Table(2): MSE of estimated exponential reliability function under the first proposed loss function. Table(5): MSE of estimated exponential reliability function under the second proposed loss function. Table(6): MSE of estimated exponential reliability function under the second proposed loss function. Table(7): MSE of estimated exponential reliability function under the second proposed loss function. Table(8): MSE of estimated exponential reliability function under the second proposed loss function.
and (10) in Appendix-E.