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Two-sample order-r U-statistic and first projections

Definition

Fix $m, n \geq r$ and independent samples $X_{1}, \dots, X_{m} \sim P$ and $Y_{1}, \dots, Y_{n} \sim Q$ , each IID within its group. For a kernel from Two-sample U-statistic parameter, define

$U_{m, n} : = \frac{1}{(\binom{m}{r}) (\binom{n}{r})} \sum_{I \in (\binom{{1, \dots, m}}{r})} \sum_{J \in (\binom{{1, \dots, n}}{r})} h (X_{I}; Y_{J}) .$

Here $I, J$ are unordered index subsets, and $h (X_{I}; Y_{J})$ means that the selected observations are listed in increasing index order in their respective blocks. Block symmetry makes the value independent of that ordering. Under $E ∣ h ∣^{2} < \infty$ for the $P^{\otimes r} \otimes Q^{\otimes r}$ law, write

$g_{X} (X_{1}) : = E [h (X_{1}, X_{2}, \dots, X_{r}; Y_{1}, \dots, Y_{r}) ∣ X_{1}] - θ (P, Q; h),$ $g_{Y} (Y_{1}) : = E [h (X_{1}, \dots, X_{r}; Y_{1}, Y_{2}, \dots, Y_{r}) ∣ Y_{1}] - θ (P, Q; h) .$

These are the centered first projections under $P$ and $Q$ ; they are square-integrable equivalence classes, so changes on their respective null sets do not change any later claim.

Remarks

The displayed conditional-expectation notation is shorthand for the completed section-integral versions under the product laws. The corresponding functions are measurable and square-integrable by First projections are measurable and square-integrable ↗.

The finite sum defining $U_{m, n}$ is a measurable real statistic: the finite sample-vector coordinate maps are measurable by Coordinate maps and permutations are measurable on finite products, composition with $h$ is measurable, and its finite sum and constant scaling remain measurable by Arithmetic and lattice operations preserve measurability whenever they are defined. This is a statistic in the sense of Statistics and estimators.

Depends on

Used by

Studentized two-sample U-statistic permutation test Algorithm
Positive kernel second moment does not imply nondegenerate first projections Counterexample
The weak null does not center every mixed kernel assignment Counterexample
Studentized two-sample U-statistic and permutation distribution Definition
A studentized rank U-statistic under a weak null Example
Studentized difference of means with equal means and unequal variances Example
First projections are measurable and square-integrable Lemma
Mixture coupling for permuted two-sample U-statistics Lemma
Pooled-mixture projection and variance Lemma
Two-sample Hoeffding decomposition Lemma
Permutation CLT for a two-sample U-statistic under a weak null Theorem
Projection CLT for a two-sample U-statistic Theorem

Dependency tree · two levels

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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, §3 (standard reference, not scraped)