Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(1): 343-358

Online first article November 3, 2016      Printed January 31, 2017

https://doi.org/10.4134/BKMS.b160139

Copyright © The Korean Mathematical Society.

Berry-Esseen bounds of recursive kernel estimator of density under strong mixing assumptions

Yu-Xiao Liu and Si-Li Niu

Henan University of Urban Construction, Tongji University

Abstract

Let $\{X_i\}$ be a sequence of stationary $\a$-mixing random variables with probability density function $f(x)$. The recursive kernel estimators of $f(x)$ are defined by $$ \widehat{f}_n(x)=\frac{1}{n\sqrt{b_n}}\sum^n_{j=1}b_j^{-\frac{1}{2}}K\Big(\frac{x-X_j}{b_j}\Big)~~\mbox{and}~~ \widetilde{f}_n(x)=\frac{1}{n}\sum^n_{j=1}\frac{1}{b_j}K\Big(\frac{x-X_j}{b_j}\Big), $$ where $0

Keywords: Berry-Esseen bound, recursive kernel estimator, $\alpha$-mixing

MSC numbers: 62G07, 62G20

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