Sharp Lower Bound on the Minimax Risk for Multinomial Uniformity Testing via a Conditional Central Limit Theorem
2026-07-06 • Information Theory
Information Theory
AI summaryⓘ
The authors study how well one can test if data is uniformly distributed across categories when considering a specific measurement scale (ℓ_p) and a large number of categories and samples. They focus on a middle ground where the problem is neither too easy nor too hard, and they analyze the minimax risk, which measures the best worst-case testing error. Previously, an upper bound on this risk was known from a version of the problem using Poisson distributions. The authors prove a matching lower bound by using advanced probability tools, thereby exactly characterizing the minimax risk for the original multinomial problem.
minimax testinggoodness-of-fituniformitymultinomial distributionℓ_p normPoisson mixturecentral limit theoremrisk boundsignal-to-noise ratiostatistical hypothesis testing
Authors
Alon Kipnis
Abstract
We study minimax goodness-of-fit testing for uniformity from $n$ multinomial observations over $N$ categories against $\ell_p$ departures of size $ε_n$. Writing $u_n:=ε_n^2 n\,N^{3/2-2/p}/\sqrt{2}$ for the associated signal-to-noise ratio, we focus on the intermediate regime $N=o(n^2)$ with $u_n\to u^*\in(0,\infty)$, in which the minimax risk converges to a nontrivial constant. In the Poissonized version of the problem this constant equals $2Φ(-u^*/2)$ \cite{Kipnis2025minimax}, yielding an upper bound on the multinomial minimax risk. Here we prove the matching lower bound. The key step is a conditional central limit theorem for weighted sums under a Poisson mixture prior, conditioned on the total count. Together with the upper bound in \cite{Kipnis2025minimax}, this gives an exact sharp-constant characterization of the multinomial minimax risk in the intermediate regime.