  For exponentially distributed lifetimes, we consider the prediction of future
order statistics based on having observed the first $m$ order statistics. We
focus on the previously less explored aspects of predicting: (i) an arbitrary
pair of future order statistics such as the next and last ones, as well as (ii)
the next $N$ future order statistics. We provide explicit and exact Bayesian
credible regions associated with Gamma priors, and constructed by identifying a
region with a given credibility $1-\lambda$ under the Bayesian predictive
density. For (ii), the HPD region is obtained, while a two-step algorithm is
given for (i). The predictive distributions are represented as mixtures of
bivariate Pareto distributions, as well as multivariate Pareto distributions.
For the non-informative prior density choice, we demonstrate that a resulting
Bayesian credible region has matching frequentist coverage probability, and
that the resulting predictive density possesses the optimality properties of
best invariance and minimaxity.
