Conditional limit laws for goodness-of-fit tests

TL;DR

This paper investigates the asymptotic behavior of goodness-of-fit statistics in exponential family models, demonstrating their approximation by parametric bootstrap distributions and providing theoretical expansions for Rao-Blackwell estimates.

Contribution

It introduces a new understanding of the conditional distribution of goodness-of-fit tests, showing their closeness to bootstrap distributions in large samples, and derives uniform Edgeworth expansions for Rao-Blackwell estimates.

Findings

Conditional distribution approximates bootstrap distribution in large samples

Provides uniform Edgeworth expansions for Rao-Blackwell estimates

Enhances understanding of goodness-of-fit tests in exponential families

Abstract

We study the conditional distribution of goodness of fit statistics of the Cram\'{e}r--von Mises type given the complete sufficient statistics in testing for exponential family models. We show that this distribution is close, in large samples, to that given by parametric bootstrapping, namely, the unconditional distribution of the statistic under the value of the parameter given by the maximum likelihood estimate. As part of the proof, we give uniform Edgeworth expansions of Rao--Blackwell estimates in these models.