Bayesian Prediction Regions and Density Estimation With Type-2 Censored Data

For exponentially distributed lifetimes, we consider the prediction of future order statistics based on having observed the first <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula>-order statistics. We focus on the previously less explored aspects of predicting: 1) an arbitrary pair of future order statistics, such as the next and last ones, as well as 2) the next <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1-\lambda$</tex-math></inline-formula> under the Bayesian predictive density. For (2), the highest posterior density region is obtained, while a two-step algorithm is given for (1). The predictive distributions are represented as mixtures of bivariate Pareto distributions, as well as multivariate Pareto distributions. For the noninformative 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.