On Estimation of Stress-Strength Reliability Using Lower Record Values from Proportional Reversed Hazard Family

Abstract

In this article, a study on the stress-strength parameter $P=P(X>Y)$ based on lower record values from one parameter proportional reversed hazard family (PRHF) has been conducted. The classical and Bayesian results of Khan and Arshad (UMVU Estimation of Reliability Function and Stress-Strength Reliability from Proportional Reversed Hazard Family Based on Lower Records. American Journal of Mathematical and Management Sciences, 35(2), 171–181) and Condino et al. (Likelihood and Bayesian estimation of P(Y < X) using lower record values from a general class of distributions. Statistical Papers), when the strength $X$ and stress $Y$ variables belong to different family of distributions from PRHF have been generalized. Uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE) and Bayes estimator (BE) are obtained for the powers of the parameter and reliability functions $R\left(t\right)=P(X>t)$ and $P.$ The estimators of three parametric functions, namely, powers of parameter, $R(t)$ and $P$ are interrelated, whereas, in the literature, researchers have handled the three estimation problems separately. Moreover, it is has been shown that the expressions for $R(t)$ and $P$ are not required to estimate them. In this article, the technique of obtaining estimators of $R(t)$ and $P$ is simpler as it does not require Rao-Blackwellization. Simulation studies have been performed for analyzing the behavior of the proposed estimators. An example using real data has also been considered as an illustration.

KEYWORDS AND PHRASES:

• Bayes estimator
• lower records
• maximum likelihood estimator
• proportional reversed hazard family
• reliability
• stress-strength reliability
• uniformly minimum variance unbiased estimator

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F10
• 62N05

Acknowledgement

The authors would like to thank the editor and the reviewers for their comments which led to considerable improvement in this article.