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Bejleri Valbona,Sartore Luca,Nandram Balgobin 한국통계학회 2022 Journal of the Korean Statistical Society Vol.51 No.3
Bayesian prediction limits are constructed based on some maximum allowed probability of wrong prediction. However, the frequency of wrong prediction in a long run often exceeds this probability. The literature on frequentist and Bayesian prediction limits, and their interpretation is sparse; more attention is given to prediction intervals obtained based on parameter estimates or empirical studies. Under the Poisson distribution, we investigate frequentist properties of Bayesian prediction limits derived from conjugate priors. The frequency of wrong prediction is used as a criterion for their comparison. Bayesian prediction based on the uniform and Jeffreys’ non-informative priors yield one sided prediction limits that can be interpreted in a frequentist context. It is shown here, by proving a theorem, that Bayesian lower prediction limit derived from Jeffreys’ noninformative prior is the only optimal (largest) Bayesian lower prediction limit that possesses frequentist properties. In addition, it is concluded as corollary that there is no prior distribution such that Bayesian upper and lower prediction limits obtained from it will both coincide with their respective frequentist prediction limits. Our results are based on asymptotic considerations. An example with real data is included, and the sensitivity of the Bayesian prediction limits with respect to conjugate priors is numerically explored through simulations.
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