Kaplan-Meier V- and U-statistics

2018/10/11 by Fernández, Tamara, Rivera, Nicolás
#FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1810.04806

Abstract

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

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