Kaplan-Meier V- and U-statistics

In this paper, we study Kaplan-Meier V- and U-statistics respectively defined as theta((F) over cap (n)) = Sigma(i,j) K(X-[i:n], X-[j:n]) and theta(U)((F) over cap (n)) = Sigma(i not equal j) K(X-[i:n], X-[j:n]) WiWj/ Sigma(i not equal j) W(i)W(j )where (F) over cap (n) is the Kaplan-Meier estimator, {W-1, ..., W-n} are the Kaplan-Meier weights and K : (0, infinity)(2) -> R is a symmetric kernel. As in the canonical setting of uncensored data, we differentiate between two asymptotic behaviours for theta((F) over cap (n)) and theta(U) ((F) over cap (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.