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December, 1991 Central Limit Theorems for $L_p$ Distances of Kernel Estimators of Densities Under Random Censorship
Miklos Csorgo, Edit Gombay, Lajos Horvath
Ann. Statist. 19(4): 1813-1831 (December, 1991). DOI: 10.1214/aos/1176348372


A sequence of independent nonnegative random variables with common distribution function $F$ is censored on the right by another sequence of independent identically distributed random variables. These two sequences are also assumed to be independent. We estimate the density function $f$ of $F$ by a sequence of kernel estimators $f_n(t) = (\int^\infty_{-\infty}K((t - x)/h(n))d\hat{F}_n(x))/h(n),$ where $h(n)$ is a sequence of numbers, $K$ is kernel density function and $\hat{F}_n$ is the product-limit estimator of $F.$ We prove central limit theorems for $\int^T_0|f_n(t) - f(t)|^p d\mu(t), 1 \leq p < \infty, 0 < T \leq \infty,$ where $\mu$ is a measure on the Borel sets of the real line. The result is tested in Monte Carlo trials and applied for goodness of fit.


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Miklos Csorgo. Edit Gombay. Lajos Horvath. "Central Limit Theorems for $L_p$ Distances of Kernel Estimators of Densities Under Random Censorship." Ann. Statist. 19 (4) 1813 - 1831, December, 1991.


Published: December, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0747.60022
MathSciNet: MR1135150
Digital Object Identifier: 10.1214/aos/1176348372

Primary: 60F05
Secondary: 60G15 , 62G10

Keywords: $L_1$ norm , Censored data , Kernel estimator , strong approximation , Wiener process

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 4 • December, 1991
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