On the UMVUE and Closed-Form Bayes Estimator for Pr(X<Y<Z) and its Generalizations
Abstract
This article considers the parametric estimation of Pr(X<Y<Z) and its generalizations based on several well-known one-parameter and two-parameter continuous distributions. It is shown that for some one-parameter distributions and when there is a common known parameter in some two-parameter distributions, the uniformly minimum variance unbiased estimator can be expressed as a linear combination of the Appell hypergeometric function of the first type, F1 and the hypergeometric functions 2F1 and 3F2. The Bayes estimator based on conjugate gamma priors and Jefferys' non-informative priors under the squared error loss function is also given as a linear combination of 2F1 and F1. Alternatively, a convergent infinite series form of the Bayes estimator involving the F1 function is also proposed. In model generalizations and extensions, it is further shown that the UMVUE can be expressed as a linear combination of a Lauricella series, FD(n), and the generalized hypergeometric function, pFq, which are generalizations of F1 and 2F1 respectively. The generalized closed-form Bayes estimator is also given as a convergent infinite series involving FD(n). To gauge the performances of the UMVUE and the closed-form Bayes estimator for P against other well-known estimators, maximum likelihood estimates, Lindley approximation estimates and Markov Chain Monte Carlo estimates for P are also computed. Additionally, asymptotic confidence intervals and Bayesian highest probability density credible intervals are also constructed.