Bayesian prediction regions and density estimation with type-2 censored data

TL;DR

This paper develops Bayesian prediction regions and density estimators for future order statistics in exponentially distributed data with type-2 censoring, providing explicit formulas and demonstrating optimality and coverage properties.

Contribution

It introduces explicit Bayesian credible regions for predicting future order statistics under gamma priors, including the HPD region, with exact formulas and optimality analysis.

Findings

Bayesian credible regions have matching frequentist coverage.

Predictive densities are mixtures of Pareto distributions.

Optimality properties like invariance and minimaxity are demonstrated.

Abstract

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.