Statistical Distances in Goodness-of-fit

Statistical distances or divergences have a long history in the scientific literature, where they are used for a variety of purposes, including that of testing for goodness of fit. In the present work, we discuss the role of distances or divergences in the context of model selection via testing. Specifically, we construct a goodness of fit test for testing simple null hypotheses and study the asymptotic distribution of the test statistic under the null. We obtain a locally quadratic representation of the test statistic and exemplify the derived results in the case of testing for normality. To do this, we identify the kernel that enters the local quadratic representation of the test statistic, obtain the asymptotic distribution of the test statistic, and illustrate its performance via simulation.