The Annals of Statistics

Transfer of tail information in censored regression models

Ingrid Van Keilegom and Michael G. Akritas

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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

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

First available in Project Euclid: 23 September 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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