🤖 AI Summary
While existing distribution testing theory has established the order of sample complexity, it lacks a precise characterization of the practical power differences among methods sharing the same asymptotic order, leaving the optimal choice ambiguous in practice. This work presents the first systematic study of sharp constants in distribution testing, focusing on uniformity and identity testing over large alphabets. By combining large-sample asymptotics under total variation distance, minimax risk bounds, and hypothesis testing theory with constant-level precision, we uncover the critical role sharp constants play in determining test power, signal-to-noise ratio, and optimal binning parameter selection. Analogous to Fisher information and Pinsker’s constant in estimation theory, our results provide a rigorous theoretical foundation for selecting testing procedures and tuning binning strategies, substantially enhancing empirical testing performance.
📝 Abstract
Distribution goodness-of-fit testing has developed a powerful rate-level theory: we often know how the required sample size scales with the alphabet size, the separation from the null, and the target error probability. Uniformity testing is the canonical example. One can distinguish the uniform distribution on $N$ categories from alternatives at total-variation distance at least $ε$ with far fewer than $N$ samples, and the optimal scaling is now well understood.
But rate-level theory leaves an important question unresolved: among several tests with the same sample-complexity order, which one actually gives the best risk or power? This is a constant-level question. It is especially relevant in modern applications where distribution testing is used not merely as an asymptotic abstraction, but as a practical design tool.
This note argues that sharp constants in distribution testing play a role analogous to Fisher information in parametric estimation and Pinsker's constant in nonparametric estimation. First, they distinguish between tests that are all rate-optimal but not equally powerful. Second, they reveal the effective signal-to-noise ratio governing the testing problem. Third, they can guide tuning-parameter choices in downstream applications. We illustrate this perspective through large-alphabet uniformity testing and then explain why the same logic matters for choosing the number of bins in calibration testing.