Kaplan-Meier V- and U-statistics

Abstract

In this paper, we study Kaplan-Meier V- and U-statistics respectively defined as $\theta (\widehat{F}_{n})=\sum _{i,j}K(X_{[i:n]},X_{[j:n]})W_{i}W_{j}$ and $\theta _{U}(\widehat{F}_{n})=\sum _{i\neq j}K(X_{[i:n]},X_{[j:n]})W_{i}W_{j}/\sum _{i\neq j}W_{i}W_{j}$, 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.