Kaplan-Meier V- and U-statistics

Abstract

In this paper, we study Kaplan-Meier V- and U-statistics respectively defined as $\theta ({\hat{F}}_n)={\sum }_{i,j}K(X_{[i:n]},X_{[j:n]})W_i{W}_j$ and ${\theta }_U({\hat{F}}_n)={\sum }_{i\ne j}K(X_{[i:n]},X_{[j:n]})W_i{W}_j/{\sum }_{i\ne j}{W}_i{W}_j$, where ${\hat{F}}_n$ is the Kaplan-Meier estimator, $\{W_1,\dots ,W_n\}$ 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 $\theta ({\hat{F}}_n)$ and ${\theta }_U({\hat{F}}_n)$. 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.