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

Eman Ahmed Abdulateef,Abbas Najim Salman

doi:10.30526/32.1.1825

Eman Ahmed Abdulateef, Abbas Najim Salman

Open Access

https://doi.org/10.30526/32.1.1825

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Abstract

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.

Highlights

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
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 (α, λ) decreasing and upside-down bathtub shaped failure rates depending on shape parameter α[2]
Hassan& Basheikh in (2012), estimated R, 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]

Summary

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 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. The Maximum Likelihood estimator for the unknown shape parameters α and β will be respectively as follows [2]: α. Substitute equation (24) and (25) in equation (5), the shrinkage estimation of R(s,k) based on modified Thompson type shrinkage weight factor will be :

Simulation Study

Numerical Simulation Results