The Annals of Statistics

Central Limit Theorems for $L_p$ Distances of Kernel Estimators of Densities Under Random Censorship

Miklos Csorgo, Edit Gombay, and Lajos Horvath

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Abstract

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.

Article information

Source
Ann. Statist., Volume 19, Number 4 (1991), 1813-1831.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348372

Digital Object Identifier
doi:10.1214/aos/1176348372

Mathematical Reviews number (MathSciNet)
MR1135150

Zentralblatt MATH identifier
0747.60022

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62G10: Hypothesis testing 60G15: Gaussian processes

Keywords
Censored data kernel estimator $L_1$ norm Wiener process strong approximation

Citation

Csorgo, Miklos; Gombay, Edit; Horvath, Lajos. Central Limit Theorems for $L_p$ Distances of Kernel Estimators of Densities Under Random Censorship. Ann. Statist. 19 (1991), no. 4, 1813--1831. doi:10.1214/aos/1176348372. https://projecteuclid.org/euclid.aos/1176348372


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