ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Studentized difference of means with equal means and unequal variances

Example

Assume AC. Let $P$ assign probability $1 / 2$ to each of $- 1, 1$ , and let $Q$ assign probability $1 / 2$ to $0$ and probability $1 / 4$ to each of $- 2, 2$ . Take $h (x, y) = x - y$ , so $r = 1$ and $θ (P, Q; h) = μ_{P} - μ_{Q}$ . Let $m / N \to 1 / 4$ . For a group $W_{1}, \dots, W_{k}$ , put ${\overset{ˉ}{W}}_{k} = k^{- 1} \sum_{i = 1}^{k} W_{i}$ and ${\hat{σ}}_{k}^{2} = (k - 1)^{- 1} \sum_{i = 1}^{k} (W_{i} - {\overset{ˉ}{W}}_{k})^{2}$ for ${\hat{σ}}_{m}^{2}$ and ${\hat{σ}}_{n}^{2}$ in the studentized statistic.

Facts & Assumptions

Given: AC, the finite laws $P, Q$ , kernel $h (x, y) = x - y$ , and sample-size ratio $p = 1 / 4$ .

[F1]

Two-sample U-statistic parameter: $θ (P, Q; h) = E_{P} X - E_{Q} Y$ .

[F2]

Two-sample order-r U-statistic and first projections: for $r = 1$ , $U_{m, n} = {\overset{ˉ}{X}}_{m} - {\overset{ˉ}{Y}}_{n}$ , $g_{X} (x) = E [h (x, Y)] - θ$ , and $g_{Y} (y) = E [h (X, y)] - θ$ .

[F3]

Permutation CLT for a two-sample U-statistic under mixed-assignment centering: under the all-assignment centered kernel premise, the raw permutation limit of $\sqrt{m} U$ has variance $r^{2} σ_{\overset{ˉ}{P}}^{2} / (1 - p)$ .

[F4]

Studentized permutation validity under mixed-assignment centering: consistent direct and pooled-mixture variance estimates give standard-normal true and permutation limits.

[F5]

Moments, variance, and covariance on a probability space: variance is the expectation of the squared centered deviation.

[F6]

Chebyshev's inequality on a finite probability space: for a finite random variable, its sample-average deviation probability is bounded by its variance divided by the squared threshold.

[A1]

AC is assumed by the asymptotic permutation and sampling limit results [F3]–[F4]. The Axiom of Choice

Verification

Given: AC, $P, Q, h$ , and $p = 1 / 4$ . Verify the weak null, projection conditions, studentizer consistency, and the unstudentized variance mismatch.

1.1F1F5givenalgebra

Both means are zero. The variances are $σ_{P}^{2} = 1$ and $σ_{Q}^{2} = 2$ , so [F1] gives $θ = 0$ . The kernel is antisymmetric, and under every P/Q assignment its expectation is the difference of two zero means, hence zero. Its second moment is the sum of the two positive variances, so it is finite and strictly positive for every assignment.

1.2F5F6givenalgebra

For $k$ observations $W_{1}, \dots, W_{k}$ , the sample variance is ${\hat{σ}}_{k}^{2} = \frac{k}{k - 1} {k^{- 1} \sum_{i} W_{i}^{2} - (k^{- 1} \sum_{i} W_{i})^{2}}$ for $k \geq 2$ . Under either finite-support law $P$ or $Q$ , Chebyshev [F6] applied to the averages of $W_{i}$ and $W_{i}^{2}$ gives convergence in probability to their expectations, so ${\hat{σ}}_{m}^{2} \to 1$ and ${\hat{σ}}_{n}^{2} \to 2$ . Under IID ${\overset{ˉ}{P}}_{N}$ rows, $∣ W_{i} ∣ \leq 2$ uniformly in $N$ , so the variances of both averages are bounded by constants divided by their row sizes; [F6] again gives convergence to their row expectations. These means are $0$ and $p_{N} (1) + (1 - p_{N}) (2) \to 7 / 4$ , so both mixture-sample variance estimates converge to $7 / 4$ .

2.1F2F5step 1.1algebra

From [F2], $g_{X} (x) = x$ and $g_{Y} (y) = - y$ , so their variances are $1$ and $2$ . The pooled mixture has mean zero and variance $σ_{{\overset{ˉ}{P}}_{N}}^{2} = p_{N} σ_{P}^{2} + (1 - p_{N}) σ_{Q}^{2} = 2 - p_{N} \to \frac{7}{4} .$ It is positive.

3.1A1F4step 1.1step 2.1step 1.2

Under AC [A1], the hypotheses of [F4] now hold, including positive direct and pooled projection variances, so the studentized statistic has standard-normal true-sampling and permutation limits. This example has a weak parameter null while $P \neq Q$ .

4.1A1F2F3step 2.1algebra∎

Under AC [A1], step 1.2 and [F4] give $\sqrt{m} V_{m, n}^{S E} \to_{P} \sqrt{1 + \frac{p}{1 - p} 2} = \sqrt{5 / 3}$ and $S_{m, n} \Rightarrow N (0, 1)$ under the true sampling law. Thus $\sqrt{m} U_{m, n} = (\sqrt{m} V_{m, n}^{S E}) S_{m, n} \Rightarrow N (0, 5 / 3)$ by Slutsky. The raw permutation limit instead has variance $\frac{σ_{\overset{ˉ}{P}}^{2}}{1 - p} = \frac{7 / 4}{3 / 4} = \frac{7}{3}$ by [F3]. To specify the resulting two-sided permutation test, let ${\hat{R}}_{N} (t) = P_{π} (\sqrt{m} U_{m, n}^{π} \leq t ∣ Z)$ and $q_{N} (u) = \inf {t : {\hat{R}}_{N} (t) \geq u}$ . For $α \in (0, 1)$ , reject when $\sqrt{m} U_{m, n} < q_{N} (α / 2)$ or $\sqrt{m} U_{m, n} > q_{N} (1 - α / 2)$ . The conditional CDF convergence in [F3] to the continuous strictly increasing law $N (0, 7 / 3)$ implies $q_{N} (α / 2) \to_{P} - z_{1 - α / 2} \sqrt{7 / 3}$ and $q_{N} (1 - α / 2) \to_{P} z_{1 - α / 2} \sqrt{7 / 3}$ . Since the true-sampling statistic converges to $N (0, 5 / 3)$ and this limit has no atoms at either limiting critical value, the rejection probability tends to $P (∣ N (0, 5 / 3) ∣ > z_{1 - α / 2} \sqrt{7 / 3}) = 2 (1 - Φ (z_{1 - α / 2} \sqrt{\frac{7}{5}})) < α,$ where $z_{1 - α / 2} > 0$ is the standard-normal quantile. Thus the unstudentized equal-tail permutation test is conservative in this example, and studentization corrects this variance mismatch.

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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 3.2 and Example 3.1 (standard reference, not scraped)