The Annals of Statistics

Nearest Neighbor Estimation of a Bivariate Distribution Under Random Censoring

Michael G. Akritas

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We consider the problem of estimating the bivariate distribution of the random vector $(X, Y)$ when $Y$ may be subject to random censoring. The censoring variable $C$ is allowed to depend on $X$ but it is assumed that $Y$ and $C$ are conditionally independent given $X = x$. The estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of $Y$ given $X = x$ over a range of values of $x$. The weak convergence of the centered estimator multiplied by $n^{1/2}$ is obtained, and a closed-form expression for the covariance function of the limiting process is given. It is shown that the proposed estimator is optimal in the Beran sense. This is similar to an optimality property the product-limit estimator enjoys. Using the proposed estimator of the bivariate distribution, an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established.

Article information

Ann. Statist., Volume 22, Number 3 (1994), 1299-1327.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Conditional empirical processes conditional Kaplan-Meier estimator weak convergence Beran optimality polynomial regression


Akritas, Michael G. Nearest Neighbor Estimation of a Bivariate Distribution Under Random Censoring. Ann. Statist. 22 (1994), no. 3, 1299--1327. doi:10.1214/aos/1176325630.

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