Kaplan-Meier V and U-statistics

10/11/2018

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by   Tamara Fernández, et al.

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In this paper we study Kaplan-Meier V and U-statistics defined as θ(F̂_n)=∑_i,jK(X_[i:n],X_[j:n])W_iW_j and θ_U(F̂_n)=∑_i≠ jK(X_[i:n],X_[j:n])W_iW_j/∑_i≠ jW_iW_j, where F̂_n denotes the Kaplan-Meier estimator, W_i's are the Kaplan-Meier weights and K is a symmetric kernel. As in the canonical setting of uncensor data, we differentiate between two limit behaviours for θ(F̂_n) and θ_U(F̂_n). Additionally, we derive an asymptotic canonical V-statistic representation for the Kaplan-Meier V and U-statistics. By using this representation we study properties of the asymptotic distribution. Finally, we study the more general case θ(F̂)=∫_RK(x_1,...,x_d)∏_j=1^ddF̂_n(x_j) for d>2. Applications to hypothesis testing are given.