Inferences on Stress-Strength Reliability from Lindley Distributions

Abstract This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes. Keywords: Asymptotic distributionCredible intervalsLindley distributionMaximum likelihood estimatorPosterior analysisPrior distributionUniformly minimum variance unbiased estimatorMathematics Subject Classification: Primary 62F10, 62F12Secondary 62F40, 62F15 Acknowledgments The authors would like to thank the Associate Editor and the reviewer for their helpful comments which improved an earlier draft of the manuscript. The first author would like to thank Kuwait Foundation for the Advancement of Sciences (KFAS) for financial support. Part of the work of the third author was supported by a grant from the Department of Science and Technology, Government of India.