Estimation of R=P(Y<X) for three-parameter Weibull distribution

In this paper we consider the estimation of the stress-strength parameter R=P(Y<X), when X and Y are independent and both are three-parameter Weibull distributions with the common shape and location parameters but different scale parameters. It is observed that the maximum likelihood estimators do not exist in this case, and we propose a modified maximum likelihood estimator, and also an approximate modified maximum likelihood estimator of R. We obtain the asymptotic distribution of the modified maximum likelihood estimators of the unknown parameters and it can be used to construct the confidence interval of R. Analyses of two data sets have also been presented for illustrative purposes.