Goodness-of-fit (GoF) testing is ubiquitous in statistics, with direct ties to model selection, confidence interval construction, conditional independence testing, and multiple testing. While testing the GoF of a simple null hypothesis provides an analyst great flexibility in the choice of test statistic while still ensuring validity, most GoF tests for composite null hypotheses are far more constrained, as the test statistic must have a tractable distribution over the entire null model space. A notable exception is co-sufficient sampling (CSS), which resamples. But CSS testing requires the null model to have a compact (in an information-theoretic sense) sufficient statistic, which only holds for a very limited class of models; even for a null model as simple as logistic regression, CSS testing is powerless. In this work, we leverage the concept of approximate sufficiency to generalize CSS testing to essentially any parametric model with an asymptotically-efficient estimator; we call our extension "approximate CSS" (aCSS) testing. We quantify the finite-sample Type I error inflation of aCSS testing and show that it is vanishing under standard maximum likelihood asymptotics, for any choice of test statistic. We also apply our proposed procedure both theoretically and in simulation to a number of models of interest.
This work is joint with Lucas Janson.