The Annals of Statistics

Transfer of tail information in censored regression models

Ingrid Van Keilegom and Michael G. Akritas

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Abstract

Consider a heteroscedastic regression model $Y = m(X) + \sigma(X)\varepsilon$, where the functions $m$ and $\sigma$ are “smooth,” and $\varepsilon$ is independent of $X$. The response variable $Y$ is subject to random censoring, but it is assumed that there exists a region of the covariate $X$ where the censoring of $Y$ is “light.” Under this condition, it is shown that the assumed nonparametric regression model can be used to transfer tail information from regions of light censoring to regions of heavy censoring. Crucial for this transfer is the estimator of the distribution of $\varepsilon$ based on nonparametric regression residuals, whose weak convergence is obtained. The idea of transferrring tail information is applied to the estimation of the conditional distribution of $Y$ given $X = x$ with information on the upper tail “borrowed ” from the region of light censoring, and to the estimation of the bivariate distribution $P(X \leq x, Y \leq y)$ with no regions of undefined mass. The weak convergence of the two estimators is obtained. By-products of this investigation include the uniform consistency of the conditional Kaplan–Meier estimator and its derivative, the location and scale estimators and the estimators of their derivatives.

Article information

Source
Ann. Statist., Volume 27, Number 5 (1999), 1745-1784.

Dates
First available in Project Euclid: 23 September 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1017939150

Digital Object Identifier
doi:10.1214/aos/1017939150

Mathematical Reviews number (MathSciNet)
MR2001b:62082

Zentralblatt MATH identifier
0957.62034

Subjects
Primary: 62G05: Estimation 62G30: Order statistics; empirical distribution functions 62H12: Estimation 62J05: Linear regression

Keywords
Asymptotic representation bivariate distribution conditional distribution nonparametric regression residuals residual distribution right censoring weak convergence

Citation

Van Keilegom, Ingrid; Akritas, Michael G. Transfer of tail information in censored regression models. Ann. Statist. 27 (1999), no. 5, 1745--1784. doi:10.1214/aos/1017939150. https://projecteuclid.org/euclid.aos/1017939150


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