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Consistency transfers from IID mixture samples to permuted groups

Statement

Assume AC. Let $P, Q$ be probability laws and let $m_{N} + n_{N} = N$ , with $p_{N} = m_{N} / N \to p \in (0, 1)$ . Write ${\overset{ˉ}{P}}_{N} = p_{N} P + (1 - p_{N}) Q$ . For each $k_{N} \in {m_{N}, n_{N}}$ , let $T_{N}$ be a measurable statistic of $k_{N}$ observations and let $c_{N}$ be a deterministic target. If

$T_{N} (V_{1}, \dots, V_{k_{N}}) - c_{N} \to 0 in probability when V_{i} are IID {\overset{ˉ}{P}}_{N},$

then $T_{N}$ applied to either group of a uniform random permutation of the pooled independent sample $X_{1}, \dots, X_{m_{N}} \sim P$ , $Y_{1}, \dots, Y_{n_{N}} \sim Q$ also converges in probability to $c_{N}$ . The statement does not require $p_{N} - p = O (N^{- 1 / 2})$ because the IID comparison law uses the exact weight $p_{N}$ .

Facts & Assumptions

Given: Independent IID samples from $P$ and $Q$ , a uniform permutation of their pooled $N$ observations, and an estimator/statistic consistent for $c_{N}$ on IID ${\overset{ˉ}{P}}_{N}$ samples of the same group size.

[F1]

Hypergeometric and binomial composition counts are contiguous: the hypergeometric label count in a size- $k_{N}$ permuted group is contiguous to the binomial label count in a size- $k_{N}$ IID mixture sample.

[F2]

Identical distribution and IID families: each source group is IID with its stated common law.

[F3]

Independent random elements: the source groups and the random subset selection are independent as specified.

[F4]

Statistics and estimators: a statistic is a measurable function of the ordered observations; an estimator is such a statistic with a specified target.

[A1]

AC is inherited from the likelihood-ratio contiguity result [F1]. The Axiom of Choice

Proof

Given: The assumptions in the Statement. Prove consistency for both permuted group sizes.

1.1F1F2F3

Fix $k_{N}$ equal to either $m_{N}$ or $n_{N}$ . In an IID sample from ${\overset{ˉ}{P}}_{N}$ , the number $B_{N}$ of observations whose mixture component is $P$ has law $Bin (k_{N}, p_{N})$ . In a uniformly chosen size- $k_{N}$ group from the pooled sample with exactly $m_{N}$ $P$ -observations among $N$ , the corresponding count $H_{N}$ is hypergeometric.

1.2F2F3F4

Conditional on either count being $j$ , the ordered group has the same distribution: its $P$ -positions are uniform among the $j$ -subsets of the $k_{N}$ positions, the values at those positions are IID $P$ , and the remaining values are IID $Q$ . Therefore there is a function $f_{N} (j) \in [0, 1]$ giving the conditional probability that $∣ T_{N} - c_{N} ∣ > ε$ under either sampling scheme.

2.1A1F1step 1.2algebra

The IID mixture consistency assumption gives $E [f_{N} (B_{N})] \to 0$ . Markov's inequality implies $f_{N} (B_{N}) \to 0$ in probability under the binomial law. By contiguity [F1], $f_{N} (H_{N}) \to 0$ in probability under the hypergeometric law. Since $0 \leq f_{N} \leq 1$ , this also gives $E [f_{N} (H_{N})] \to 0$ . AC is used through [F1] only.

3.1step 1.1step 1.2step 2.1∎

By step 1.2, $E [f_{N} (H_{N})]$ is exactly the probability that the statistic differs from $c_{N}$ by more than $ε$ for the randomly permuted size- $k_{N}$ group. This tends to zero. The argument applies separately to $k_{N} = m_{N}$ and $k_{N} = n_{N}$ , proving the assertion.

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Sources

E. Y. Chung and J. P. Romano, Exact and Asymptotically Robust Permutation Tests, Stanford Technical Report 2011-05, Lemma 3.3 and proof (standard reference, not scraped)