Locally sharp goodness-of-fit testing in sup norm for high-dimensional counts

We consider testing the goodness-of-fit of a distribution against alternatives separated in sup norm. We study the twin settings of Poisson-generated count data with a large number of categories and high-dimensional multinomials. In previous studies of different separation metrics, it has been found that the local minimax separation rate exhibits substantial heterogeneity and is a complicated function of the null distribution; the rate-optimal test requires careful tailoring to the null. In the setting of sup norm, this remains the case and we establish that the local minimax separation rate is determined by the finer decay behavior of the category rates. The upper bound is obtained by a test involving the sample maximum, and the lower bound argument involves reducing the original heteroskedastic null to an auxiliary homoskedastic null determined by the decay of the rates. Further, in a particular asymptotic setup, the sharp constants are identified.