🤖 AI Summary
This work investigates the minimax detection boundary for uniformity testing of multinomial distributions under ℓₚ deviations, focusing on the intermediate regime where the sample size satisfies \( N = o(n^2) \) and the signal-to-noise ratio converges to a finite positive constant. By introducing a Poissonized model, constructing a Poisson mixture prior, and applying a conditional central limit theorem for weighted sums, the authors establish—for the first time—a matching lower bound on the minimax risk in this regime. This lower bound precisely coincides with the known upper bound, thereby fully characterizing the asymptotic behavior of the minimax risk at the level of sharp constants: the risk converges to the nontrivial constant \( 2\Phi(-u^*/2) \).
📝 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.