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We consider the problem of goodness-of-fit testing for a model that has atleast one unknown parameter that cannot be eliminated by transformation.Examples of such problems can be as simple as testing whether a sample consistsof independent Gamma observations, or whether a sample consists of independentGeneralised Pareto observations given a threshold. Over time the approach todetermining the distribution of a test statistic for such a problem has movedtowards on-the-fly calculation post observing a sample. Modern approachesinclude the parametric bootstrap and posterior predictive checks. We argue thatthese approaches are merely approximations to integrating over the posteriorpredictive distribution that flows naturally from a given model. Further, weattempt to demonstrate that shortcomings which may be present in the parametricbootstrap, especially in small samples, can be reduced through the use ofobjective Bayes techniques, in order to more reliably produce a test with thecorrect size.

On Determining the Distribution of a Goodness-of-Fit Test Statistic