$V$-statistics and Variance Estimation
This work addresses a theoretical gap in statistical inference for ensemble methods like random forests, offering incremental improvements in variance estimation.
This paper tackles the problem of analyzing asymptotics of V-statistics when kernel size grows with sample size, demonstrating asymptotic normality under regularity conditions and providing a unified variance estimation method for U- and V-statistics, with an empirically more accurate estimator.
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.