Some asymptotic properties of kernel estimators of a density function in case of censored data
type: Item
instance of: scholarly article
title: Some asymptotic properties of kernel estimators of a density function in case of censored data
zbMATH Open document ID: 0603.62047
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: The author considers a random censorship model based on nonnegative independent random variables X and Y with densities f and g resp.. Given a sample \(Z_ i=\min (X_ i,Y_ i)\), \(\delta_ i=\chi (X_ i\leq Y_ i)\), \(1\leq i\leq n\), he investigates the following density estimator for \(f\): \[ \hat f_ n(x)=(1/h)\int_{{\mathbb{R}}}K((x-y)/h)dF_ n^{(KM)}(y), \] where K is a density itself and \(F_ n^{(KM)}\) is the Kaplan-Meier estimator for \(F_ X\). [See also \textit{J. Blum} and \textit{V. Susarla}, Multivariate analysis V, Proc. 5th int. Symp., Pittsburgh 1978, 213-222 (1980; Zbl 0462.62030) or \textit{A. Földes}, \textit{L. Rejtö} and \textit{B. B. Winter}, Period. Math. Hung. 12, 15-29 (1981; Zbl 0461.62038).] Four types of bandwidths h are considered, a deterministic sequence \(h_ n\), respectively a sequence \(h_{n_ 1}\), where \(n_ 1=\#\) of uncensored data, and N.N.-bandwidths \(R(n)\), \(R(n_ 1)\) resp. It is shown that in all cases \(\hat f_ n\) can be approximated by kernel density estimators based on appropriate empirical distributions and the well known limit theorems for those apply and result in e.g. strong uniform consistency and asymptotic normality of \(\hat f_ n\) (in Cor. (2,ii) read \(h_ n=o(n^{-1/5})\) instead of \(O(n^{-1/3}))\). It turns out that the variances of the N.N.-estimates are censoring free.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 3974076
zbMATH Keywords: nearest neighbor bandwidths
zbMATH Keywords: random censorship model
zbMATH Keywords: Kaplan-Meier estimator
zbMATH Keywords: kernel density estimators
zbMATH Keywords: strong uniform consistency
zbMATH Keywords: asymptotic normality
author: Jan Mielniczuk
reviewed by: Ulrich Stadtmüller
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349954
OpenAlex ID: W1969507337
DOI: 10.1214/AOS/1176349954
description: scientific article; zbMATH DE number 3974076