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http://dx.doi.org/10.4134/BKMS.b160139

BERRY-ESSEEN BOUNDS OF RECURSIVE KERNEL ESTIMATOR OF DENSITY UNDER STRONG MIXING ASSUMPTIONS  

Liu, Yu-Xiao (School of Mathematics and Physics Henan University of Urban Construction)
Niu, Si-Li (Department of Mathematics Tongji University)
Publication Information
Bulletin of the Korean Mathematical Society / v.54, no.1, 2017 , pp. 343-358 More about this Journal
Abstract
Let {$X_i$} be a sequence of stationary ${\alpha}-mixing$ random variables with probability density function f(x). The recursive kernel estimators of f(x) are defined by $$\hat{f}_n(x)={\frac{1}{n\sqrt{b_n}}{\sum_{j=1}^{n}}b_j{^{-\frac{1}{2}}K(\frac{x-X_j}{b_j})\;and\;{\tilde{f}}_n(x)={\frac{1}{n}}{\sum_{j=1}^{n}}{\frac{1}{b_j}}K(\frac{x-X_j}{b_j})$$, where 0 < $b_n{\rightarrow}0$ is bandwith and K is some kernel function. Under appropriate conditions, we establish the Berry-Esseen bounds for these estimators of f(x), which show the convergence rates of asymptotic normality of the estimators.
Keywords
Berry-Esseen bound; recursive kernel estimator; ${\alpha}-mixing$;
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