Bayes Optimal Goodness-of-fit Tests
In this talk, Bayesian procedures are introduced into goodness-of-fit tests to give, on the alternative, a band of distributions around the tested distribution, and it is assumed that the statistician wishes to detect alternatives in this band. Neyman-Pearson theory is used to lead to the optimal test procedure. The procedure is quite general and is explored for testing fit for distributions with both known and unknown parameters. It is shown that the standard quadratic goodness-of-fit tests of Cramer-von~Mises family are approximately Bayes optimal.
A particular use of Bayesian methods allows consideration of the problem of testing the distribution of latent variables when these are connected by a known relationship to a set of observed variables. The technique is used to advance an interesting procedure introduced in Geology by Krumbein and for a modern example, to test the distribution of the frailty term (random effects) in a Cox Proportional Hazards model.