🤖 AI Summary
This paper addresses the goodness-of-fit testing problem for multiclass Poisson count data against the uniform distribution, where the alternative hypothesis consists of uniform Poisson rate sequences lying outside an ℓₚ-ball (p ≤ 2) of radius ε. We characterize the asymptotic minimax risk of this testing problem at constant accuracy for the first time. We propose an adaptive test: it employs collision-based statistics in the small-sample regime and smoothly transitions to the chi-square test in the large-sample regime. Our analysis integrates collision statistic theory, asymptotic normality of linear statistics, optimization over N-dimensional sequences, and analogy with the Poisson–Gaussian white noise model—yielding risk bounds tight up to constant factors. Both theoretical analysis and empirical experiments demonstrate that the proposed method significantly outperforms both the standard chi-square test and pure collision-based tests in finite-sample settings, while delivering highly accurate asymptotic risk estimates.
📝 Abstract
We study the problem of testing the goodness of fit of occurrences of items from many categories to a Poisson distribution uniform over the categories, against a class of alternative hypotheses obtained by the removal of an $ell_p$ ball, $p leq 2$, of radius $epsilon$ around the sequence of uniform Poisson rates. We characterize the minimax risk for this problem as the expected number of samples $n$ and the number of categories $N$ go to infinity. Our result enables the comparison at the constant level of the many estimators previously proposed for this problem, rather than at the rate of convergence of the risk or the scaling order of the sample complexity. The minimax test relies exclusively on collisions in the small sample limit but behaves like the chisquared test otherwise. Empirical studies over a range of problem parameters show that the asymptotic risk estimate is accurate in finite samples and that the minimax test is significantly better than the chisquared test or a test that only uses collisions. Our analysis involves the reduction to a structured subset of alternatives, establishing asymptotic normality for linear statistics, and solving an optimization problem over $N$-dimensional sequences that parallels classical results from Gaussian white noise models.