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Pooled-mixture projection and variance

Statement

Use the setup of Two-sample order-r U-statistic and first projections and suppose $h$ is antisymmetric under exchange of its two blocks and has finite second moment under every independent P/Q assignment to its $2 r$ positions. For $t \in (0, 1)$ put ${\overset{ˉ}{P}}_{t} = t P + (1 - t) Q$ and define

$g_{t} (z) = \int h (z, z_{2}, \dots, z_{r}; w_{1}, \dots, w_{r}) d {\overset{ˉ}{P}}_{t}^{\otimes (2 r - 1)} - θ ({\overset{ˉ}{P}}_{t}, {\overset{ˉ}{P}}_{t}; h) .$

Then $θ ({\overset{ˉ}{P}}_{t}, {\overset{ˉ}{P}}_{t}; h) = 0$ , the first projection in the second block is $- g_{t}$ , and $σ_{t}^{2} : = E_{{\overset{ˉ}{P}}_{t}} [g_{t}^{2}]$ is finite. The map $t \mapsto σ_{t}^{2}$ is continuous on $(0, 1)$ . In particular, with $N = m + n$ and $p_{N} = m / N \to p \in (0, 1)$ ,

$r^{2} (1 + \frac{m}{n}) σ_{p_{N}}^{2} ⟶ \frac{r^{2}}{1 - p} σ_{p}^{2} .$

Facts & Assumptions

Given: A block-symmetric kernel, its antisymmetry under exchange of blocks, and finite second moments under all mixed P/Q assignments.

[F1]

Two-sample order-r U-statistic and first projections defines the block kernel and its first projections.

[F2]

First projections are measurable and square-integrable gives the section-integral versions and their L2 bounds.

[F3]

Moments, variance, and covariance on a probability space defines the second-moment variance $E [g_{t}^{2}]$ when $E g_{t} = 0$ .

[F4]

Cauchy-Schwarz inequality for $L^{2}$ makes cross products of the finitely many square-integrable mixed-assignment section functions integrable.

[F5]

Fubini's theorem for L^1 functions on a sigma-finite product supplies measurable integrable section functions and equality of iterated integrals on the mixed product probability spaces.

Proof

Given: The hypotheses in the Statement. Prove the pooled mean, sign relation, finiteness, and continuity claims.

1.1F1givenalgebra

Under ${\overset{ˉ}{P}}_{t}^{\otimes 2 r}$ , swapping the first and second blocks preserves the product law and changes $h$ to $- h$ . Therefore $θ ({\overset{ˉ}{P}}_{t}, {\overset{ˉ}{P}}_{t}; h) = - θ ({\overset{ˉ}{P}}_{t}, {\overset{ˉ}{P}}_{t}; h)$ , so it is zero.

1.2F2F3F4F5givenalgebra

Expand each factor ${\overset{ˉ}{P}}_{t} = t P + (1 - t) Q$ in the section integral. This writes $g_{t}$ as a finite sum $\sum_{η} c_{η} (t) f_{η} (z)$ , where $η$ ranges over assignments of P and Q to the remaining $2 r - 1$ coordinates, $c_{η} (t)$ is a polynomial, and $f_{η}$ is the corresponding section integral. For either first-coordinate law $R \in {P, Q}$ , write $E_{R, η}$ for product expectation with independent coordinates having law $R$ first and the laws specified by $η$ in the remaining positions. The mixed-moment assumption gives both $E_{P, η} [h^{2}] < \infty$ and $E_{Q, η} [h^{2}] < \infty$ . Cauchy--Schwarz [F4] also makes $h$ integrable with first-coordinate law $(P + Q) / 2$ . Fubini [F5] on that product space supplies one measurable version of $f_{η}$ , valid $(P + Q)$ -almost everywhere after setting it to zero on the common exceptional set. Cauchy--Schwarz on the remaining probability product then gives $\int ∣ f_{η} ∣^{2} d R \leq E_{R, η} [h^{2}] < \infty$ for each $R \in {P, Q}$ , as in the section-integral argument of [F2]. Consequently every cross integral $\int f_{η} f_{η^{'}} d R$ is finite by [F4]. Expanding $E_{{\overset{ˉ}{P}}_{t}} g_{t}^{2} = t E_{P} [g_{t}^{2}] + (1 - t) E_{Q} [g_{t}^{2}]$ now gives a finite sum of polynomial coefficients times these finite cross integrals, so it is finite and continuous in $t$ .

2.1F1F2step 1.1algebra

Holding the first coordinate of $h$ fixed and integrating the other $2 r - 1$ coordinates with law ${\overset{ˉ}{P}}_{t}$ defines $g_{t}$ , since the kernel mean in [F1] is zero by step 1.1. For the projection in the second block, exchange the blocks inside the integral. Antisymmetry changes the sign and the common product law is unchanged, so the second-block projection is $- g_{t}$ .

3.1step 1.2algebra∎

Since $m / n = p_{N} / (1 - p_{N})$ and $p_{N} \to p$ , step 1.2 gives $σ_{p_{N}}^{2} \to σ_{p}^{2}$ , while $1 + m / n = 1 / (1 - p_{N}) \to 1 / (1 - p)$ . Multiplication by $r^{2}$ proves the displayed limit.

Depends on

Used by

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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 Remark 3.1 (standard reference, not scraped)