$V$-statistics and Variance Estimation

TL;DR

This paper introduces a comprehensive framework for analyzing the asymptotic behavior of V-statistics, extending existing results to cases where kernel size grows with sample size, and offers improved variance estimation methods.

Contribution

It provides a novel reduction of V-statistics to U-statistics for asymptotic analysis and proposes a more accurate variance estimator applicable to both types.

Findings

Asymptotic normality holds when kernel size grows with sample size.

Unified variance estimation method for U- and V-statistics.

Application to ensemble methods like random forests.

Abstract

This paper develops a general framework for analyzing asymptotics of $V$-statistics. Previous literature on limiting distribution mainly focuses on the cases when $n \to \infty$ with fixed kernel size $k$. Under some regularity conditions, we demonstrate asymptotic normality when $k$ grows with $n$ by utilizing existing results for $U$-statistics. The key in our approach lies in a mathematical reduction to $U$-statistics by designing an equivalent kernel for $V$-statistics. We also provide a unified treatment on variance estimation for both $U$- and $V$-statistics by observing connections to existing methods and proposing an empirically more accurate estimator. Ensemble methods such as random forests, where multiple base learners are trained and aggregated for prediction purposes, serve as a running example throughout the paper because they are a natural and flexible application of $V$-statistics.