## The Annals of Statistics

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

#### 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

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
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