Bulletin of the Korean Mathematical Society (대한수학회보)

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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)
  • Received : 2016.02.18
  • Published : 2017.01.31
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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

Acknowledgement

Supported by : National Natural Science Foundation of China

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