Dominance of posterior predictive densities over plug-in densities for order statistics in exponential distributions

Many researchers have proposed numerous Bayesian predictive densities for Type-II censored data that is generated by ordered observations. However, their evaluations of predictive densities were insufficient because the Bayesian predictive density includes prior parameters, thus we suffer from the selection of the prior parameters. In this study, we consider two types of predictive densities, posterior predictive and plug-in, for observations from an exponential distribution of Type-II censored data. We discuss a suitable predictive density using the risk with the Kullback-Leibler loss function. In our setting, we consider a Gamma prior, which is a conjugate prior for mathematical tractability. We prove that the posterior predictive density with an improper Gamma prior provides the dominance of the posterior predictive density over the plug-in densities without depending on the selection of an unknown parameter in our setting. Finally, we show that the posterior predictive density outperforms the plug-in densities in terms of coverage probabilities for unobserved data by censoring in a simulation study.