Kaplan-meier v-and u-statistics

Tamara Fernández; Nicolás Rivera

doi:10.1214/20-EJS1704

Kaplan-meier v-and u-statistics

Tamara Fernández
, Nicolás Rivera

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	1872-1916
Number of pages	45
Journal	Electronic Journal of Statistics
Volume	14
Issue number	1
DOIs	https://doi.org/10.1214/20-EJS1704
State	Published - 2020
Externally published	Yes

Keywords

Kaplan-Meier estimator
Right-censoring
V-statistics

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title = "Kaplan-meier v-and u-statistics",

abstract = "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])WiWj and θU {\^(}Fn)=i≠j K(X[i:n],X[j:n])WiWj /i≠j WiWj,where{\^ }Fn is the Kaplan-Meier estimator, \{W1,…,Wn\} 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{\^(}Fn)andθU {\^(}Fn). 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.",

keywords = "Kaplan-Meier estimator, Right-censoring, V-statistics",

author = "Tamara Fern{\'a}ndez and Nicol{\'a}s Rivera",

year = "2020",

doi = "10.1214/20-EJS1704",

language = "English",

volume = "14",

pages = "1872--1916",

journal = "Electronic Journal of Statistics",

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publisher = "Institute of Mathematical Statistics",

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TY - JOUR

T1 - Kaplan-meier v-and u-statistics

AU - Fernández, Tamara

AU - Rivera, Nicolás

PY - 2020

Y1 - 2020

N2 - 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])WiWj and θU (̂Fn)=i≠j K(X[i:n],X[j:n])WiWj /i≠j WiWj,where ̂Fn is the Kaplan-Meier estimator, {W1,…,Wn} 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(̂Fn)andθU (̂Fn). 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.

AB - 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])WiWj and θU (̂Fn)=i≠j K(X[i:n],X[j:n])WiWj /i≠j WiWj,where ̂Fn is the Kaplan-Meier estimator, {W1,…,Wn} 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(̂Fn)andθU (̂Fn). 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.

KW - Kaplan-Meier estimator

KW - Right-censoring

KW - V-statistics

UR - https://www.scopus.com/pages/publications/85088927740

U2 - 10.1214/20-EJS1704

DO - 10.1214/20-EJS1704

M3 - Article

AN - SCOPUS:85088927740

SN - 1935-7524

VL - 14

SP - 1872

EP - 1916

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

IS - 1

ER -

Kaplan-meier v-and u-statistics

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Keywords

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