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Permutation CLT for a two-sample U-statistic under a weak null

Statement

Assume AC. Let $X_{1}, \dots, X_{m}$ be IID $P$ and independently let $Y_{1}, \dots, Y_{n}$ be IID $Q$ , with $N = m + n$ , $m / N \to p \in (0, 1)$ , and $θ (P, Q; h) = 0$ . Fix $r$ and a measurable kernel $h$ that is symmetric in each block and antisymmetric under block exchange. Assume for every assignment of independent P- and Q-distributed arguments to the $2 r$ positions that $0 < E [h^{2}] < \infty$ . Let $σ_{p}^{2} = E_{\overset{ˉ}{P}} [g_{\overset{ˉ}{P}}^{2}] > 0$ for $\overset{ˉ}{P} = p P + (1 - p) Q$ , where $g_{\overset{ˉ}{P}}$ is its first projection. Then the conditional permutation CDF of $\sqrt{m} U_{m, n}$ satisfies

$\sup_{t \in R} ∣ {\hat{R}}_{m, n}^{\sqrt{m} U} (t) - Φ ⁣ (\frac{t}{τ_{p}}) ∣ \to_{P} 0, τ_{p}^{2} = \frac{r^{2}}{1 - p} σ_{p}^{2} .$

Facts & Assumptions

Given: The sample and kernel hypotheses in the Statement, including finite positive second moments under every mixed argument assignment.

[F1]

Two-sample order-r U-statistic and first projections defines $U_{m, n}$ and its two first projections.

[F2]

Two-sample Hoeffding decomposition gives the first-order projection plus the finite sum of canonical higher-order terms.

[F3]

Second-moment order of the degenerate remainder makes the scaled higher-order remainder negligible.

[F4]

Pooled-mixture projection and variance identifies the pooled projection $g_{\overset{ˉ}{P}}$ and its variance scale $τ_{p}^{2}$ .

[F5]

Conditional CLT for a random subset of an IID finite population gives conditional normal convergence for a uniform sample without replacement from an IID pooled row.

[F6]

Mixture coupling for permuted two-sample U-statistics couples the original permuted U-statistic to the IID-mixture permuted U-statistic under antisymmetry and all mixed second moments.

[A1]

AC is used through [F5]. The Axiom of Choice

Proof

Given: The weak-null and mixed-moment kernel hypotheses and $m / N \to p \in (0, 1)$ . Prove the conditional permutation CDF limit for $\sqrt{m} U_{m, n}$ .

1.1F1F2F3F4given

Let $p_{N} = m / N$ and ${\overset{ˉ}{P}}_{N} = p_{N} P + (1 - p_{N}) Q$ . Under an IID ${\overset{ˉ}{P}}_{N}$ row, antisymmetry gives pooled parameter zero and first projections $g_{N}$ and $- g_{N}$ by [F4]. Expanding the $2 r$ mixture factors shows that $E_{{\overset{ˉ}{P}}_{N}^{\otimes 2 r}} h^{2}$ is a finite convex combination of the mixed-assignment second moments in the hypotheses, hence is uniformly bounded in $N$ . The same bound applies to every mixed assignment when both row laws are ${\overset{ˉ}{P}}_{N}$ , so the triangular version of [F3] applies to the two groups formed by a uniform permutation and makes $\sqrt{m}$ times the degenerate remainder tend to zero in joint probability.

2.1F1F4F5givenalgebra

The linear projection for a uniform size- $m$ subset $A$ of the pooled row is $\sqrt{m} L_{N}^{π} = \frac{r N}{n \sqrt{m}} (\sum_{i \in A} g_{N} ({\overset{ˉ}{Z}}_{i}) - \frac{m}{N} \sum_{i = 1}^{N} g_{N} ({\overset{ˉ}{Z}}_{i})) .$ Expanding each factor of ${\overset{ˉ}{P}}_{N}$ shows that $g_{N}$ is a finite linear combination of the section-integral functions for the finitely many mixed P/Q assignments. The assignment-wise second-moment bounds make these functions square-integrable under P and Q; their coefficients are bounded as $p_{N} \to p$ . If $H$ is the sum of their squared absolute values, then $H$ is integrable under both P and Q and $g_{N}^{2} \leq C H$ for one constant $C$ . Hence, for $K > 0$ , $E_{{\overset{ˉ}{P}}_{N}} [g_{N}^{2} 1_{{g_{N}^{2} > K}}] \leq C (E_{P} [H 1_{{H > K / C}}] + E_{Q} [H 1_{{H > K / C}}]) ⟶ 0,$ so the row $g_{N} ({\overset{ˉ}{Z}}_{i})$ has uniformly integrable squares. Its variance is $σ_{N}^{2} \to σ_{p}^{2}$ . Let $S_{g, N}^{2} = (N - 1)^{- 1} \sum_{i = 1}^{N} (g_{N} ({\overset{ˉ}{Z}}_{i}) - {\overset{ˉ}{g}}_{N})^{2}$ be its empirical finite-population variance. Apply step 1.1 of [F5] to this IID row: the uniformly integrable squares give $N^{- 1} \sum_{i} g_{N} ({\overset{ˉ}{Z}}_{i})^{2} - σ_{N}^{2} \to_{P} 0$ , while $E {\overset{ˉ}{g}}_{N} = 0$ and $Var ({\overset{ˉ}{g}}_{N}) = σ_{N}^{2} / N \to 0$ . Therefore $S_{g, N}^{2} - σ_{N}^{2} \to_{P} 0$ and $S_{g, N}^{2} \to_{P} σ_{p}^{2}$ .

3.1A1F5step 1.1step 2.1algebra

Put $T_{N}^{π} = \frac{\sum_{i \in A} (g_{N} ({\overset{ˉ}{Z}}_{i}) - {\overset{ˉ}{g}}_{N})}{\sqrt{m n S_{g, N}^{2} / N}}$ when $S_{g, N}^{2} > 0$ , and set it to zero otherwise. Apply [F5] to this random subset. The exact identity on ${S_{g, N}^{2} > 0}$ is $\sqrt{m} L_{N}^{π} = c_{N} T_{N}^{π}$ , where $c_{N} = r \sqrt{N / n} S_{g, N}$ . Since $S_{g, N}^{2} \to_{P} σ_{p}^{2} > 0$ , $c_{N} \to_{P} r σ_{p} / \sqrt{1 - p} = τ_{p}$ and the event $S_{g, N}^{2} = 0$ has probability tending to zero. The conditional CDF of $T_{N}^{π}$ converges uniformly in probability to $Φ$ by [F5]; thus, for each fixed $t$ , the conditional CDF of $c_{N} T_{N}^{π}$ converges in probability to $Φ (t / τ_{p})$ . Indeed, on $c_{N} > 0$ its conditional CDF is $F_{N, \overset{ˉ}{Z}} (t / c_{N})$ , whose difference from $Φ (t / c_{N})$ is bounded by $\sup_{u} ∣ F_{N, \overset{ˉ}{Z}} (u) - Φ (u) ∣$ , and $Φ (t / c_{N}) \to_{P} Φ (t / τ_{p})$ . The zero-variance event is negligible. The conditional probability of the remainder exceeding any fixed threshold tends to zero in probability, since its expectation tends to zero by step 1.1 and Markov's inequality. Another threshold squeeze and continuity of the normal CDF give the same pointwise conditional limit for $\sqrt{m} U_{m, n} ({\overset{ˉ}{Z}}^{π})$ .

4.1F6step 3.1algebra

Let $π, π^{'}$ be conditionally independent uniform permutations. By step 3.1, the two mixture-row statistics have asymptotically independent $N (0, τ_{p}^{2})$ limits: conditionally on $\overset{ˉ}{Z}$ , their joint CDF is the product of the two conditional CDFs, and each factor converges in probability to a constant. Apply the coupling [F6] to both permutations; the union bound and its $L^{2}$ conclusion preserve this joint limit for the original sample.

5.1step 3.1step 4.1algebra∎

Write $R_{N} (t)$ for the original conditional permutation CDF. The one-permutation convergence gives $E R_{N} (t) \to Φ (t / τ_{p})$ ; the two independent permutations in step 4.1 give $E [R_{N} (t)^{2}] \to Φ (t / τ_{p})^{2}$ . Therefore $Var (R_{N} (t)) \to 0$ , so $R_{N} (t) \to_{P} Φ (t / τ_{p})$ at every $t$ . Since the limit CDF is continuous, choose finite tail cutoffs and a finite grid with arbitrarily small $Φ$ increments; monotonicity bounds the supremum between grid values. This proves the stated uniform convergence.

Remarks

The proof uses the exact triangular mixture and the locally strengthened coupling estimate. It requires neither zero kernel means for other assignments nor a convergence rate for $m / N - p$ . The weak null is needed for the accompanying true-sampling limit; antisymmetry alone centers the permutation distribution.

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Sources

E. Y. Chung and J. P. Romano, Asymptotically Valid and Exact Permutation Tests Based on Two-Sample U-Statistics, Stanford Technical Report 2011-09, Theorem 3.1 and Appendix A, Lemmas A.1–A.2 (standard reference, not scraped)