Estimation of a New Stress Strength Index for One Parameter Exponential Family

Abstract

Stress-strength reliability is widely used in manufacturing industry for producing good quality equipment. A new stress-strength index has been introduced when strengths of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> independent components follow exponential distributions with different scale parameters. We obtain the maximum likelihood estimator, a uniformly minimum variance unbiased estimator, a Bayes estimator, and an analogue of the best scale equivariant estimator for the new index. It is shown that the Bayes estimators' limit is a generalized Bayes estimator under the squared error loss function. The asymptotic distribution of the MLE is derived using the multivariate delta method. A detailed comparison of the risk performance of all these estimators is done numerically. The results are illustrated by two real data analyses.