On Shrinkage Estimation for R ( s , k ) in Case of Exponentiated Pareto Distribution

This paper concerns with deriving and estimating the reliability of the multicomponent system in stress-strength model R(s,k), when the stress and strength are identical independent distribution (iid), follows two parameters Exponentiated Pareto Distribution (EPD) with the unknown shape and known scale parameters. Shrinkage estimation method including Maximum likelihood estimator (MLE), has been considered. Comparisons among the proposed estimators were made depending on simulation based on mean squared error (MSE) criteria.


Introduction
The reliability of the multicomponent stress-strength model (s out of k) system, denoted by R(s,k)refers to the system functioning when at minimums (1≤s≤k) of components survive.In other words, this system works well if at least s out of k components resist the stress.Bhattacharyya and Johnson in (1974) was the first who studied and derived R(s,k) [1].Noted that, when s=1 and s=k is respectively referring to parallel and series systems.The model mentioned used in many applications in physics and engineering and many authors had studied and estimated R(s,k)for example: Afify in (2010), showed that the Exponentiated Pareto distribution denoted by EP (α, λ) used quite successfully in studying many lifetime data and the EP (, λ) decreasing and upside-down bathtub shaped failure rates depending on shape parameter α [2].Hassan& Basheikh in (2012), estimated  , using Bayes and non-Bayes estimation methods when the strength and stress are non-identical and follows the Exponentiated Pareto distribution [3], Rao et al in (2016), estimated the reliability system in a multicomponent stress-strength when stress and strength follows Exponentiated Weibull distribution for different shape parameters [4], and in (2017) Abbas and Fatima, they estimated the reliability of the multicomponent system in stress-strength model for Exponentiated Weibull distribution, using; ML, MOM and the conclude results approved that the Shrinkage estimator using Shrinkage weight function was the best [5].
In this paper we estimate R(s,k) based on Exponentiated Pareto distribution EP(α, λ) with unknown shape parameter α and known scale parameter λ using several shrinkage estimation methods depends on (MLE) methods and make a comparison of the considered estimation methods through Monte Carlo simulation via mean squared error (MSE) criteria.
It is well known, the EP (α, λ) is a special case of Exponentiated Lomax distribution (ELD), when ( 1 where the CDF of ELD has the form below [6]. Let X be a random variable follows two parameters Exponentiated Pareto distribution EPD (α, λ).The probability density function (p.d. f.) of X will be.
Here,  refers to shape parameter and  refers to scale parameter [2].Implies, the cumulative distribution function (CDF) of X as below:  , ,  1 1   0; ,  0 (3) Assume  ,  , … ,  are strength random variable follows EP(α,λ) and subject to common stress random variable Y which is distributed as EP(β,λ) .Then, the reliability system for a multicomponent in stress-strength model R(s,k) will be [1]: R(s,k) =P(at least s of the X1, X2,…, Xkexceed Y) Now we estimation methods of R(s,k) by the following:

Maximum Likelihood Estimator (MLE)
Suppose the strength random sample be of size n say  ,  , … ,  follow EPD (,  and  ,  , … … ,  be the stress random sample of size m follow EPD (,  The Maximum Likelihood function for the observed sample is given as:  =L (, ; , ) =∏   ∏   ) (6) From equation (2) and the equation (6) become Take Logarithm to both sides, we get: Thus, the Maximum Likelihood estimator for the unknown shape parameters  and β will be respectively as follows [2]: By substitute Equations ( 8) and (9) in equation ( 5), we obtain the maximum likelihood estimator for  , as below:

Shrinkage Estimation Method (Sh)
As Thompson suggested in (1968), the shrinkage estimator of α denoted by (α ) is defined as below: α ∅ α α 1 ∅ α α (13) He shrinks a usual estimator  to prior information using shrinkage weight factor ∅ α and he believed  is closed to α [5], [7].We apply the unbiased estimator  as a usual estimator and   as a prior information of α in this paper.
Thus, the shrinkage estimator of the shape parameterα of EP (α,λ) will be as follows:  ∅   1 ∅   (14) And the same way, the shrinkage estimator for the shape parameter β of EP(β,λ) will be as the following:

The Shrinkage Weight Function (Sh1):
In this subsection, the shrinkage weight factor will be considered as a function of sizes n and m respectively and taking the forms below: ∅  |sin /| , and ∅  |sin /| where, n and m refer to the sample size of X and Y. Therefore, the shrinkage estimator using shrinkage weight function of  and β which is defined in equations ( 14) and ( 15 Then, the estimation of , which is defined in Equation ( 5) using shrinkage weight function will be:-

Constant Shrinkage Weight Function (Sh2)
We suggest in this subsection constant shrinkage weight factor ∅  0.1, and∅  0.1.Therefore, the shrinkage estimator using specific constant weight factor will be as follows:  0.1  0.9  (19)  0.1  0.9  (20) Substitute equation ( 19) and (20) in equation ( 5) to obtain the shrinkage estimation of R(s,k) using the above constant shrinkage weight factor as below: ; k, i, j are integers (21)

Modified Thompson Type Shrinkage Weight Function (Th)
In this subsection, we modify the shrinkage weight factor of Thompson type estimator as below, [7]   *0.001 * 0.001 where Var (α ) = and Var ( ) = Therefore, the shrinkage estimator ofα and β using modified shrinkage weight factor are respectively as below: Substitute equation ( 24) and (25) in equation ( 5), then the shrinkage estimation of R(s,k) based on modified Thompson type shrinkage weight factor will be : ; k, i, j are integers (26)

Simulation Study
In this section, numerical results were studied to compare the performance of the suggested estimators for , , using different sample size n and m = (15, 25, 50 and 100), based on 1000 replication via MSE criteria.For this purpose, Mote Carlo simulation was employed by generating the random sample from continuous uniform distribution defined on the interval (0,1) as u1,u2,…, un; v1, v2,…, vm.Transform uniform random samples to follow EPD (α, λ) using (c.d.f) as below, [8]: And by the same way, calculate  to obtain the  : Compute the real value of R(s,k) in equation ( 5) and the value of estimation methods of all suggested methods  , ,  , ,  , and  , in Equations ( 12), ( 18), ( 21) and ( 26) respectively.
Based on (L=1000) replication, we calculate the MSE for all proposed estimation methods of R(s,k) as follows: where  , refers to the proposed estimators of real value of R , .

Discussion Numerical Simulation Results
From the tables above, for all n=(15,25,50,100) and m=(15,25,50,100) we conclude that, the shrinkage estimator using Modified Thompson type shrinkage weight factor to estimate the reliability  , is the bestsince the (MSE)of RTh(s,k) was less than in the other methods.