Estimation of<tex>$rm P[Y≪X]$</tex>for Weibull Distributions

Abstract

This paper deals with the estimation of R=P[Y<X] when X, and Y are two independent Weibull distributions with different scale parameters, but having the same shape parameter. The maximum likelihood estimator, and the approximate maximum likelihood estimator of R are proposed. We obtain the asymptotic distribution of the maximum likelihood estimator of R. Based on the asymptotic distribution, the confidence interval of R can be obtained. We also propose two bootstrap confidence intervals. We consider the Bayesian estimate of R, and propose the corresponding credible interval for R. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.