Open Access
2020 Kaplan-Meier V- and U-statistics
Tamara Fernández, Nicolás Rivera
Electron. J. Statist. 14(1): 1872-1916 (2020). DOI: 10.1214/20-EJS1704


In this paper, we study Kaplan-Meier V- and U-statistics respectively defined as $\theta (\widehat{F}_{n})=\sum _{i,j}K(X_{[i:n]},X_{[j:n]})W_{i}W_{j}$ and $\theta _{U}(\widehat{F}_{n})=\sum _{i\neq j}K(X_{[i:n]},X_{[j:n]})W_{i}W_{j}/\sum _{i\neq j}W_{i}W_{j}$, where $\widehat{F}_{n}$ is the Kaplan-Meier estimator, $\{W_{1},\ldots ,W_{n}\}$ are the Kaplan-Meier weights and $K:(0,\infty )^{2}\to \mathbb{R}$ 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.


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Tamara Fernández. Nicolás Rivera. "Kaplan-Meier V- and U-statistics." Electron. J. Statist. 14 (1) 1872 - 1916, 2020.


Received: 1 December 2018; Published: 2020
First available in Project Euclid: 24 April 2020

zbMATH: 07200246
MathSciNet: MR4090787
Digital Object Identifier: 10.1214/20-EJS1704

Primary: 62G20 , 62N02
Secondary: 62G30

Keywords: Kaplan-Meier estimator , right-censoring , V-statistics

Vol.14 • No. 1 • 2020
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