Density estimation for samples satisfying a certain absolute regularity condition
type: Item
instance of: scholarly article
title: Density estimation for samples satisfying a certain absolute regularity condition
zbMATH Open document ID: 0551.62022
DOI: 10.1016/0378-3758(84)90041-7
published in: Journal of Statistical Planning and Inference
publication date: 1984-01-01T00:00:00Z
review text: Let \(\{\xi_ i\}\) be a strictly stationary sequence of random variables which are defined on a probability space (\(\Omega\),\({\mathcal A},P)\). For \(a\leq b\), let \({\mathcal M}^ b_ a\) denote the \(\sigma\)-algebra of events generated by \(\xi_ a,...,\xi_ b\). The sequence \(\{\xi_ i\}\) satisfies the absolute regularity condition if \[ \beta (n)=E\{\sup_{A\in {\mathcal M}_ n^{\infty}}| P(A| {\mathcal M}^ 0_{-\infty})-P(A)| \}\downarrow 0\quad (n\to \infty). \] Let f(x) be the density function of \(\xi_ 1\). As an estimate of f(x) the author considers \(f_ n(x)=n^{-1}\sum^{n}_{j=1}H_ n(x,\xi_ j)\), where \(H_ n(x,y)\) is a Borel-measurable function defined on a Euclidean space. In this paper asymptotic properties of the probability \(P(\sup_ x| f_ n(x)-f(x)| >\epsilon)\) are studied.
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 3878160
zbMATH Keywords: density estimation
zbMATH Keywords: empirical density
zbMATH Keywords: strictly stationary sequence
zbMATH Keywords: absolute regularity condition
zbMATH Keywords: Borel-measurable function
author: Ken-ichi Yoshihara
MaRDI profile type: Publication
full work available at URL: 0378-3758(84)90041-7
OpenAlex ID: W2021260688
cites work: A note on empirical processes of strong-mixing sequences
cites work: Q4095599
cites work: Q4057952
cites work: On Estimation of a Probability Density Function and Mode
cites work: Remarks on Some Nonparametric Estimates of a Density Function
cites work: Probability density estimation using delta sequences
cites work: Limiting behavior of U-statistics for stationary, absolutely regular processes
cites work: Probability inequalities for sums of absolutely regular processes and their applications
description: scientific article; zbMATH DE number 3878160