On the UMVUE and Closed-Form Bayes Estimator for Pr(X<Y<Z) and its Generalizations
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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 _2F_1 and _3F_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 _2F_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, _pF_q, which are generalizations of F_1 and _2F_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.
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