Kaplan-Meier V- and U-statistics

TL;DR

This paper investigates the asymptotic properties of Kaplan-Meier V- and U-statistics, deriving their asymptotic distributions and applying these results to hypothesis testing in censored data settings.

Contribution

It introduces a canonical asymptotic representation for Kaplan-Meier V- and U-statistics, extending classical results to censored data.

Findings

Derived asymptotic distributions for Kaplan-Meier V- and U-statistics.

Established a canonical V-statistic representation for these estimators.

Applied results to develop hypothesis testing methods for censored data.

Abstract

In this paper, we study Kaplan-Meier V- and U-statistics respectively defined as $\theta(\widehat{F}_n)=\sum_{i,j}K(X_{[i:n]},X_{[j:n]})W_iW_j$ and $\theta_U(\widehat{F}_n)=\sum_{i\neq j}K(X_{[i:n]},X_{[j:n]})W_iW_j/\sum_{i\neq j}W_iW_j$, where $\widehat{F}_n$ is the Kaplan-Meier estimator, $\{W_1,\ldots,W_n\}$ are the Kaplan-Meier weights and $K:(0,\infty)^2\to\mathbb R$ is a symmetric kernel. As in the canonical setting of uncensored data, we differentiate between two asymptotic behaviours for $\theta(\widehat{F}_n)$ and $\theta_U(\widehat{F}_n)$. Additionally, we derive an asymptotic canonical V-statistic representation of the Kaplan-Meier V- and U-statistics. By using this representation we study properties of the asymptotic distribution. Applications to hypothesis testing are given.