Kaplan-Meier V- and U-statistics

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 $theta_U(widehat{F}_n)=sum_{i eq j}K(X_{[i:n]},X_{[j:n]})W_iW_j/sum_{i eq j}W_iW_j$, where $widehat{F}_n$ is the Kaplan-Meier estimator, ${W_1,ldots,W_n}$ are the Kaplan-Meier weights and $K:(0,infty)^2tomathbb R$ 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.