Employ Shrinkage Estimation Technique for the Reliability System in Stress-Strength Models : special case of Exponentiated Family Distribution

A reliability system of the multi-component stress-strength model R(s,k) will be considered in the present paper ,when the stress and strength are independent and non-identically distribution have the Exponentiated Family Distribution(FED) with the unknown shape parameter α and known scale parameter λ equal to two and parameter θ equal to three. Different estimation methods of R(s,k) were introduced corresponding to Maximum likelihood and Shrinkage estimators. Comparisons among the suggested estimators were prepared depending on simulation established on mean squared error (MSE) criteria.

Weibull distribution, consuming; ML and MOM estimators and they accomplished results agreed that the shrinkage estimator was the best [7]. And in (2019), Abbas and Eman derived and estimated the formula of reliability for multicomponent in (s-s) model in case of Exponentiated Pareto distribution and they concluded that the shrinkage estimator was the best [8]. The aim of this paper is to estimate the reliability of multicomponent system in stress-strength model R(s,k) based on Family of Exponentiated distribution with unknown shape parameter α and known the parameters λ=2 and θ=3 and a=0 via different estimation methods like MLE, as well as some shrinkage methods and make comparisons among the proposed estimator methods using simulation, depending on mean squared error criteria by using special case of Family Exponentiated distribution .
The probability density function (p. d. f.) of a r.v. X following Exponentiated Family Distribution EFD has the form below [9]: Here g x; a, is the function of x and may also be of the parameter (may be vector valued) and a is constant. Furthermore, g( ; , ) refer to real valued, monotonically increasing function of with g(a; a, )=0, g(∞; a, )= ∞ and ( ; , ) refer to the derived of g ( ; , ) with respect to [9].
Consequently, the cumulative distribution function (c.d. f.) of X will be: , The first who derived the reliability of a multicomponent system in stress-strength model R(s,k) was Bhattacharyya and Johnson (1974)

Maximum Likelihood Estimator (MLE):
Consider a random sample x1,x2,…,xn of size n following EFD( , 2,3 , x(1) <x(2)<…<x(n) , the order random sample of x and y1 ,y2,…,ym is a random sample of size m follows EFD ( , 2,3 , and y(1) <y(2)<…<y(m) is the order random sample of y. Then the likelihood functions will be as follows: Thus, the maximum likelihood estimator of the parameters and β will be as follows: By substituting α , in equation (3), obtains the Maximum Likelihood estimator for R(s,k) :

Shrinkage Estimation Method (Sh1):
Thompson in 1968 assumed the problem of shrink was a usual estimator of the parameter to prior information using shrinkage weight factor ∅ ), such that 0 ∅ 1. Thompson estimating and he believed that was closed to the true value of or that may be near the true value of .Thus, the form of Thompson -Type shrinkage estimator of α say will be [10].
In this paper, we apply the unbiased estimator as a usual estimator and as a prior estimation of α in equation (7), as we mentioned, ∅ denotes the shrinkage weight factor such that 0 ∅ 1, which may be a function of ; a function of sample size (n,m) or may be constant (ad hoc basis) .

The Shrinkage Weight Function (sh1)
The shrinkage weight factor as a function of sample sizes n and m respectively will be considered in this subsection and take the form below: i.e. ∅ Beta (n,m) and ∅ Beta m, n Where n, and m are sample sizes, therefore the shrinkage estimator using shrinkage weight function of and β which is defined in equation (7), will take the following formula: Also, as Thompson mentioned, and are closed to the real value of respectively.Then, the shrinkage estimation , in equation (3) using shrinkage weight function will be:-

Constant Shrinkage Weight Factor (Sh2):
We suggest in this subsection constant shrinkage weight factor ∅ 0.05, and ∅ 0.05. Therefore, the shrinkage estimator through specific constant weight factor will be as follows: 0.05 0.95 Substitute equation (11) and (12) in equation (3) to obtain the shrinkage estimation of R(s,k) using the above constant shrinkage weight factor as below: Where , , are integers

Modified Thompson type Shrinkage Weight Factor (th):
In this paper, the shrinkage weight factor which was considered by Thompson will be modified to give the following formulas: Therefore, the shrinkage estimator of α and β using modified shrinkage weight factor are respectively as bellow: Then the Shrinkage estimation of R(s,k) in equation (3)

Simulation Experiments
In this section, numerical results were studied to compare the performance of the different estimators of reliability using different sample size = (10, 30, 50 and 100), based on 1000 replication via MSE criteria. For this purpose, Monte Carlo simulation was employed by generating the random sample from the continuous uniform distribution defined on the interval (0,1) as u1,u2,…, un; v1, v2,…, vm. Transform uniform random samples to follow some distributions special case of the Family of Exponentiated distribution using (c. d. f.), [11].
The following steps compute the real value of R(s,k) in equation (3)   Now, we use some special case of Exponentiated Family distribution and applied the Simulation to find the value of the reliability and MSE to compare with the other methods.

Discussion Simulation results
From the tables above, for n= (10,30,50,100) besides m=(10,30,50,100) we conclude that the minimum (MSE) for the estimator of system reliability R (s,k) , (i=1,2,3,4,5),held for shrinkage estimator by using shrinkage weight function .This indicates that the shrinkage reliability estimator ( , ) is the best for any n and m and follows by shrinkage estimator using modified Thomson weight factor.

Conclusion
The simulation results exhibited that the shrinkage estimator method is suitable for estimation the parameters and system reliability for the Exponentiated Family distribution especially when shrinkage weight factor as a linear combination between unbiased estimator, and prior estimate. The result estimator perform well and will be the best estimator than the other in the sense of MSE.