Prior and Posterior Predictive P-Values in the One-Sided Location Parameter Testing Problem

In a hypothesis testing problem, the classical p-values are often perceived as measurements of the degree of surprise in the data, relative to a hypothesized model. The classical p-values commonly provide a basis for rejection of a hypothesis or a model. In this paper, we develop prior predictive and posterior predictive p-values for one sided hypothesis testing for location parameter problems. We show that for many classes of prior distributions, the infimum of the prior predictive and posterior predictive p-values are equal to the classical p-value, for very general classes of distributions. The results are in spirit similar to that in Casella and Berger (1987) in terms of reconciliation of Bayesian and frequentist evidence. The results are used through many examples relating to the one sided testing problem for location parameter.