  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.
