On the UMVUE and Closed-Form Bayes Estimator for $Pr(X<Y<Z)$ and its Generalizations

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, $F_{1}$ and the hypergeometric functions $_{2}F_{1}$ and $_{3}F_{2}.$ 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 $_{2}F_{1}$ and $F_{1}.$ Alternatively, a convergent infinite series form of the Bayes estimator involving the $F_{1}$ 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, $F_{D}^{(n)},$ and the generalized hypergeometric function, $_{p}F_{q},$ which are generalizations of $F_{1}$ and $_{2}F_{1}$ respectively. The generalized closed-form Bayes estimator is also given as a convergent infinite series involving $F_{D}^{(n)}.$ 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.