Different Estimation Methods of the Stress-Strength Reliability Power Distribution

This paper deals with estimation of the reliability system in the stressstrength model of the shape parameter for the power distribution. The proposed approach has been including different estimations methods such as Maximum likelihood method, Shrinkage estimation methods, least square method and Moment method. Comparisons process had been carried out between the various employed estimation methods with using the mean square error criteria via Matlab software package.


Introduction
The power function distribution represents one of the most important distribution approach in statistical process.It can be expressed by ~(, ) .Actually, it can be considered as a simple and flexible distribution method that can be used in different applications such as the estimation of reliability for electrical components [1].This distribution has been introduced in (1967) by Malik H.J throughout studying the exact moment of power function distribution and found a precise expression for moment of power function distribution [2].Although Bayesian estimation method of parameters have been used by several statisticians and mathematical analysts, but many researchers considered the power function distribution is better than a lot of distributions such as lognormal distribution, exponential distribution and weibull distribution.The power function distribution is as a special case for person first kind distribution that represents the simplicity of the moments for power function distribution [3].In fact blue estimation method has been introduced for the estimation of the scale and location parameter from the Log-gamma distribution.Many of others researchers were presented estimations of normal distribution parameters by using likelihood function [4][5][6].
The aim of this work is to estimate the reliability system in the stress-strength model of power function distribution.

Maximum Likelihood Estimator (MLE)
The maximum likelihood estimator is most popular method because it is approximates the minimum variance unbiased [12].Let 1 ,  2 … …   be a random sample of( 1 , 1) and 1 ,  2 … …   be a random sample of ( 2 , 1) Then, the Maximum likelihood for 1 and 2 will be: Taking the logarithm of both sides, then: Taking the partial derivation for the above equations with respect to the shape parameter and then equating the results to zero, this will lead to: ̂2 in equation ( 5); the reliability estimation of stressstrength model using maximum likelihood method will become:

Moment Method (MOM)
The moment method can be considered as the most common and a simple method that have been used in estimation of the parameters.It can be summarized by equating the population moments   = (  ) as in the following [8].
When equating the sample mean E(x) and E(y) to the corresponding population mean, then From equations ( 9) and ( 10), the estimation of parameters  1 and α 2 using moment method will be written as in the following: Substituting the equations ( 11) and (12) in equation ( 5), then the estimation of stressstrength reliability with using moment method will become:

Shrinkage Estimation Method (sh) [10][13].
The shrinkage estimation method can be considered as the Bayesian approach which has depended on the prior information.The basic reasons for using the prior estimation had been introduced by Thompson in 1968.In the shrinkage estimation method the parameter was used as initial value  0 from the past and usual estimator  ̂ through consideration them by shrinkage weight factor, ∅( ̂) , 0 < ∅( ̂) < 1, which can be written as: 2.4.1 Shrinkage Weight Function (   ) [11].
In this case, the shrinkage weight factor has used as a function of n, in which: Substituting in equation ( 17), then: Substituting  ̂1ℎ 1 , ̂2ℎ 1 in equation ( 5), then the reliability estimation of stressstrength model with using shrinkage weight function will become:
In this case, the assumption of∅( ̂) = (1, , ) for the Beta shrinkage weight factor has been taken as∅( ̂1) = (1, ), and ∅( ̂2) = (1, ) and this implies to the following shrinkage estimators: 5), then the reliability estimation of stressstrength model with using Beta shrinkage factor will become as in the following:

3-Simulation Process
The simulation process has done with using different sample size such as (30, 50 and 100) and built on 1000 iteration and using the mean square error (MSE) to measure and check the performance as in the following steps: Step 1: the random sample generated for X according to the uniform distribution over the interval (0, 1) as 1 , 2 …………  and the random sample generated for Y according to the uniform distribution over the interval (0, 1), as  1 , 2 ,…………  Step 2: transforming the above power distribution with using reliability as in the following: i=1, 2, 3 ……..n Using the same procedure .then Step 3: calculating ̂1 and  ̂2 using equations ( 6) and ( 7) respectively.

Numerical Result
Some methods of goodness of fit analysis are employed here; the measurement give an indication of best method is mean square error (MSE) from tables for all.
1.For all n=(30,50,100) and m=(30,50,100) in this work for minimum mean square error (MSE) for the stress-strength reliability estimator of power function distribution after noted the mean square error in tables, the result indicates that shrinkage estimator (Sh 2 ) is the best .2. For all n=(30,50,100) and m=(30,50,100) the minimum mean square error (MSE) for the stress-strength reliability estimator of power function distribution ,we noticed that the shrinkage estimator is the best and follows by maximum likelihood estimator (MLE), moment estimator (MOM) and least square estimator (LS).3.For the various cases when (n=30 and m=30), ( 1 =1and α 2 =1) then be moment