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arxiv: 2607.05223 · v1 · pith:P3AFWOF3 · submitted 2026-07-06 · math.ST · cs.IT· math.IT· stat.TH

Sharp Lower Bound on the Minimax Risk for Multinomial Uniformity Testing via a Conditional Central Limit Theorem

Alon Kipnis This is my paper

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classification math.ST cs.ITmath.ITstat.TH
keywords minimaxboundmultinomialriskcentralciteconditionalconstant
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The pith

Exact minimax risk pinned down for multinomial uniformity testing

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves the missing lower bound that closes the exact minimax risk for testing whether multinomial data is uniform. When you draw n samples across N categories and want to detect deviations from uniformity measured in the l_p norm, the hardest possible testing problem has a risk that converges to exactly 2*Phi(-u*/2), where u* is the limiting signal-to-noise ratio. The upper bound was already known from a Poissonized version of the problem; this paper supplies the matching lower bound, proving the Poisson approximation is tight. The key technical innovation is a conditional central limit theorem: when you model the category counts as Poisson random variables with random rates drawn from a two-point mixture prior, and then condition on the total count equaling n (which recovers the multinomial), the weighted sum statistic still converges to a Gaussian. This conditional CLT is what bridges the gap between the analytically tractable Poisson model and the multinomial model the paper targets.

Core claim

The central object is a conditional central limit theorem (Theorem 2.3) for weighted sums of Poisson mixture random variables, conditioned on their total sum equaling a fixed value n. This theorem shows that, under a Lyapunov-type moment condition and a decorrelation condition, the conditional distribution of the weighted sum converges uniformly to a standard normal. Applied to the specific quadratic weights w*_m = (m - n/N)^2 - m and a two-point symmetric prior that perturbs each category probability by +/- epsilon * N^{-1/p}, this CLT shows that the Bayes risk of the likelihood ratio test in the conditioned Poisson model converges to 2*Phi(-u*/2). Since the multinomial model is exactly the

What carries the argument

Conditional central limit theorem for weighted Poisson mixture sums conditioned on their total

If this is right

  • The exact constant 2*Phi(-u*/2) provides a precise calibration benchmark: any uniformity test in this regime can be compared against this risk floor to determine how far it is from optimal.
  • The conditional CLT technique may extend to other de-Poissonization problems where one needs to transfer sharp constants from independent Poisson models to fixed-sample-size multinomial or binomial models.
  • The two-point least-favorable prior structure identifies the specific alternative distributions that are hardest to distinguish from uniformity, which could guide the design of robust testing procedures.
  • The result confirms that the Poissonization approximation loses no information about minimax risk in the intermediate regime N = o(n^2), validating a widely used simplification.

Load-bearing premise

The proof hinges on a Lyapunov condition (third moment divided by variance to the 3/2, scaled by sqrt(N), going to zero) being satisfied for the specific quadratic weights under the mixture prior. This is verified within the paper, but it requires the regime N = o(n^2) and bounded Poisson rates; if either condition fails, the conditional CLT and hence the lower bound could break down.

What would settle it

If one could construct a test achieving risk strictly below 2*Phi(-u*/2) in the specified regime, or if the Lyapunov condition failed for the quadratic weights under the two-point prior in some parameter configuration satisfying N = o(n^2), the main theorem would be contradicted.

read the original abstract

We study minimax goodness-of-fit testing for uniformity from $n$ multinomial observations over $N$ categories against $\ell_p$ departures of size $\epsilon_n$. Writing $u_n:=\epsilon_n^2 n\,N^{3/2-2/p}/\sqrt{2}$ 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 $2\Phi(-u^*/2)$ \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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 5 minor

Summary. This paper establishes the matching lower bound for the minimax risk of multinomial uniformity testing under ℓ_p departures in the intermediate regime N=o(n²) with signal-to-noise ratio u_n → u* ∈ (0,∞). Combined with the upper bound from Kipnis [2025], this yields the exact sharp constant lim R* = 2Φ(-u*/2). The proof reduces the multinomial minimax risk to a conditioned Poisson Bayes risk under a two-point least-favorable prior, approximates the likelihood ratio test by a quadratic-weight statistic, and applies a novel conditional central limit theorem (Theorem 2.3) for weighted Poisson sums conditioned on the total count.

Significance. The paper resolves the sharp constant in the intermediate SNR regime for multinomial uniformity testing, completing the picture initiated by the Poissonized analysis. The conditional CLT (Theorem 2.3) is the key technical contribution: it handles the de-Poissonization step within the Gaussian limit calculation via a three-region Fourier decomposition, which is correctly executed. The Lyapunov condition is verified for the specific quadratic weights. The result is a falsifiable, parameter-free sharp constant. The self-citation to Kipnis [2025] for the upper bound is appropriate; the lower bound is derived independently via the conditional CLT and least-favorable prior analysis, and the constant 2Φ(-u*/2) emerges from the Gaussian shift experiment rather than by definition.

major comments (2)
  1. Lemma 3.6, Eq. (3.11): The proof argues that because all moments of T̃(w_LR) and T̃(w*) exist, the difference between moments of the two standardized statistics is o(1), and then invokes a 'standard moment convergence theorem' (Anderson et al. [2010]). This step is too terse to verify. The Taylor expansion in Lemma 5.3 gives w_LR_m = (ε²N^{2-2/p}/2)(w*_m + o(e^m + λ²)), but the o(e^m + λ²) term is not shown to be uniform in m, and its contribution to the moments of T̃(w_LR) is not bounded explicitly. Since the Poisson distribution has exponential tails, this is likely tractable, but the argument as written has a gap. Please add an explicit bound showing that the remainder term contributes o(1) to the relevant moments, or at minimum to the first two moments, under the regime N=o(n²) and u_n → u*. This is load-bearing because Lemma 3.6 is used in the proof of Theorem 2.1 to replace the LRT
  2. Lemma 3.3: The proof shows that sup_{q̃ ∈ Ã_N} Pr[ψ=0|H̃(q̃)] ≤ sup_{q ∈ A_N} Pr[ψ=0|H(q)] by arguing that q̃/‖q̃‖₁ ∈ A_N(ε,p). However, the ℓ_p norm of q̃/‖q̃‖₁ need not be ≥ ε even if ‖q̃ - q_unif‖_p ≥ ε, because normalization can shrink the departure. Specifically, if ‖q̃‖₁ > 1, then q̃/‖q̃‖₁ is closer to uniform in ℓ_p. The claim that q̃^(k)/‖q̃^(k)‖₁ ∈ A_N(ε,p) needs justification; at minimum, one should show that the ℓ_p separation is preserved up to a 1+o(1) factor under the asymptotic regime, or restrict Ã_N to sequences with ‖q̃‖₁ = 1+o(1). This step is load-bearing for the equality R* = R̃*|S_n.
minor comments (5)
  1. Section 3.1: The reference '(??)' appears twice, indicating a broken cross-reference to the Poissonized model. Please fix.
  2. Eq. (2.5): The proportionality constant for w*_m is not specified. While it does not affect the test (since scaling is absorbed into the threshold), stating the normalization explicitly would improve clarity.
  3. Lemma 3.4: The condition 'εN^{-1/p} ≤ 1/N' seems restrictive; for typical p>1 this requires ε ≤ N^{1/p-1}, which is the non-emptiness condition for A_N. Please clarify whether this is always satisfied in the asymptotic regime or is an additional constraint.
  4. Proof of Lemma 5.3: The bound on the maximal binomial coefficient as o(e^m) via Stirling is correct but the o(s²(e^m+λ²)) notation conflates the remainder across different terms. A brief clarification of which terms contribute to the remainder would help.
  5. The notation Δ_n is used in the proof of Lemma 3.7 (p.14) without being defined at that point; it is defined as nεN^{-1/p} inline. A forward reference or earlier definition would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two genuine gaps in the proofs of Lemmas 3.6 and 3.3. Both points are well-taken and require revisions. We address each below.

read point-by-point responses

  1. Referee: Lemma 3.6, Eq. (3.11): The proof argues that because all moments of T̃(w_LR) and T̃(w*) exist, the difference between moments of the two standardized statistics is o(1), and then invokes a 'standard moment convergence theorem' (Anderson et al. [2010]). This step is too terse to verify. The Taylor expansion in Lemma 5.3 gives w_LR_m = (ε²N^{2-2/p}/2)(w*_m + o(e^m + λ²)), but the o(e^m + λ²) term is not shown to be uniform in m, and its contribution to the moments of T̃(w_LR) is not bounded explicitly. Since the Poisson distribution has exponential tails, this is likely tractable, but the argument as written has a gap. Please add an explicit bound showing that the remainder term contributes o(1) to the relevant moments, or at minimum to the first two moments, under the regime N=o(n²) and u_n → u*.

    Authors: The referee is correct that the proof of Lemma 3.6 as written is too terse and contains a genuine gap. The invocation of the moment convergence theorem is not justified without an explicit bound on the contribution of the remainder term to the moments of the standardized statistics. We will revise the proof to fill this gap. The key observations are as follows. Under the prior π*, each Q_i takes values in {1/N ± εN^{-1/p}}, so the Poisson rates Λ_i = nQ_i are bounded by a universal constant C (as verified in the proof of Lemma 3.7, using u_n → u* which implies Δ_n = O(1)). For a Poisson random variable X with bounded rate λ ≤ C, we have E[e^{tX}] < ∞ for all t > 0, and in particular E[e^X] ≤ exp(C(e-1)). From the Taylor expansion in Lemma 5.3, the remainder satisfies |w^{LR}_m - (ε²N^{2-2/p}/2)w*_m| ≤ C' ε²N^{2-2/p}(e^m + λ²) for a universal constant C', uniformly in m (this uniformity follows from the explicit form of the remainder in the Taylor expansion of g_{m,λ}(s), which involves binomial coefficients bounded by 2^m/√m and polynomial terms in m and λ). Since the Poisson rates are bounded, E[e^{O_i}] is uniformly bounded, and the contribution of the remainder to E[T̃(w_{LR})] and Var[T̃(w_{LR})] is bounded by C'' ε²N^{2-2/p} · N · O(1) = O(ε²N^{3-2/p}). Under the SNR parametrization u_n = ε²nN^{3/2-2/p}/√2 with u_n → u*, we have ε²N^{3-2/p} = O(u_n²/n) = O(1/n) → 0. After standardization (dividing by the standard deviation σ_0 = √(2n²/N)), the contribution of the remainder to the first two moments of the standardized statistic is O(1/(n · σ_0)) = o(1). This explicit calculation justifies the moment convergence argument. We will include this detailed bound in the revised manuscript. revision: yes

  2. Referee: Lemma 3.3: The proof shows that sup_{q̃ ∈ Ã_N} Pr[ψ=0|H̃(q̃)] ≤ sup_{q ∈ A_N} Pr[ψ=0|H(q)] by arguing that q̃/‖q̃‖₁ ∈ A_N(ε,p). However, the ℓ_p norm of q̃/‖q̃‖₁ need not be ≥ ε even if ‖q̃ - q_unif‖_p ≥ ε, because normalization can shrink the departure. Specifically, if ‖q̃‖₁ > 1, then q̃/‖q̃‖₁ is closer to uniform in ℓ_p. The claim that q̃^(k)/‖q̃^(k)‖₁ ∈ A_N(ε,p) needs justification; at minimum, one should show that the ℓ_p separation is preserved up to a 1+o(1) factor under the asymptotic regime, or restrict Ã_N to sequences with ‖q̃‖₁ = 1+o(1). This step is load-bearing for the equality R* = R̃*|S_n.

    Authors: The referee is correct that the claim q̃/‖q̃‖₁ ∈ A_N(ε,p) does not follow from q̃ ∈ Ã_N(ε,p) without additional argument, since normalization can shrink the ℓ_p departure from uniform. We will revise the proof of Lemma 3.3 to address this. The fix is to restrict Ã_N(ε,p) to rate vectors satisfying ‖q̃‖₁ = 1 + o(1), and to show that this restriction does not change the asymptotic minimax risk. Specifically, we modify the definition of Ã_N(ε,p) to Ã_N^{(1)}(ε,p) := {q̃ ∈ [0,∞)^N : ‖q̃ - q_unif‖_p ≥ ε, ‖q̃‖₁ = 1}. With this restriction, q̃/‖q̃‖₁ = q̃ ∈ A_N(ε,p) directly, and Lemma 3.3 holds with equality R* = R̃*|S_n without gap. The key question is whether this restriction affects the lower bound. Since the least-favorable prior π* of (3.6) is supported on vectors with ‖Q‖₁ = 1 (each component is 1/N ± εN^{-1/p}, summing to 1), the Bayes risk calculation in Lemma 3.1 and the subsequent arguments are unaffected. The reduction in Lemma 3.1 only requires that the prior places mass on Ã_N, and π* places all its mass on vectors with unit sum. Therefore, the lower bound R* ≥ 2Φ(-u*/2) goes through unchanged with the restricted alternative set. For the upper bound (2.2), which comes from Kipnis [2025] in the Poissonized setting, the restriction to ‖q̃‖₁ = 1 only makes the alternative smaller, so the upper bound remains valid. We will update the definition of Ã_N and the proof of Lemma 3.3 accordingly, and add a remark explaining that the restriction to unit-sum rate vectors is without loss of generality for the asymptotic minimax risk because the least-favorable prior is supported on such vectors. revision: yes

Circularity Check

0 steps flagged

No significant circularity found

full rationale

The paper's central result (Theorem 2.1) is a lower bound on the minimax risk: lim inf R* >= 2Phi(-u*/2). The derivation chain is self-contained and does not reduce to its inputs by construction. The constant 2Phi(-u*/2) emerges from a Gaussian shift experiment (Lemma 3.7, Eq. 5.21), where the conditional CLT (Theorem 2.3) shows the test statistic converges to N(0,1) under the null and N(u*,1) under the alternative. The minimax risk of this Gaussian shift problem is a standard, independently computable quantity (2Phi(-u*/2)), not a fitted parameter renamed as a prediction. The upper bound (Eq. 2.2) is cited from Kipnis [2025] (a self-citation), but it is used only to establish the matching upper bound in Corollary 2.2; the lower bound proved here is derived independently via the conditional CLT, least-favorable prior construction (Eq. 3.6), likelihood ratio analysis (Lemma 3.5), and Lyapunov condition verification (Lemma 3.7). The self-citation provides the complementary half of the result, not the load-bearing premise for the lower bound. The conditional CLT (Theorem 2.3) is proved from scratch in Section 4.2 using characteristic function methods and a three-region decomposition (Lemma 4.1), with no circular dependency on the target result. No fitted input is called a prediction, no definition is circular, and no ansatz is smuggled via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

No free parameters are fitted; the constant emerges from the SNR definition. No new entities are postulated. The axioms are standard probability results.

axioms (3)
  • standard math Standard properties of multinomial and Poisson distributions
    Used throughout for likelihood ratios and variance calculations.
  • standard math Lyapunov condition for triangular arrays
    Assumption (iii) of Theorem 2.3, verified in Lemma 3.7 for the specific weights used.
  • standard math Lattice local CLT techniques
    The three-region decomposition in Lemma 4.1 follows Petrov [1975] and Janson [2001].

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