{"ID":5937001,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-09T15:22:02.48613467Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05223","arxiv_id":"2607.05223","title":"Sharp Lower Bound on the Minimax Risk for Multinomial Uniformity Testing via a Conditional Central Limit Theorem","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.","short_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 $u_n\\to u^*\\in(0,\\infty)$, i...","url_abs":"https://arxiv.org/abs/2607.05223","url_pdf":"https://arxiv.org/pdf/2607.05223v1","authors":"[\"Alon Kipnis\"]","published":"2026-07-06T15:37:38Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"cs.IT\"]","methods":"[]","has_code":false}