Kaplan-meier v-and u-statistics

In this paper, we study∑ Kaplan-Meier V-and U-statistics re-spectively ∑ defined as θ(̂Fⁿ)=∑_i,j K(X_[i:n],X_[j:n])WⁱW^j and θ^U (̂Fⁿ)=_i≠j K(X_[i:n],X_[j:n])W_iW_j /_i≠j W_iW_j,where ̂F_n is the Kaplan-Meier estimator, {W₁,…,W_n} are the Kaplan-Meier weights and K:(0, ∞)² → R is a symmetric kernel. As in the canonical setting of uncensored data, we differentiate between two asymptotic behaviours for θ(̂F_n)andθ_U (̂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.