Berry–Esseen bounds for density estimates under NA assumption

Abstract

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

$$\hat{f_n}(x)=\frac{1}{n\sqrt{b_n}}\sum^n_{j=1}b_j^{-\frac{1}{2}}K \left(\frac{x-X_j}{b_j}\right),\quad \tilde{f_n}(x)=\frac{1}{n}\sum^n_{j=1}\frac{1}{b_j}K \left(\frac{x-X_j}{b_j}\right)$$

and the Rosenblatt–Parzen’s kernel estimator of f(x) is defined by ${{f_n}(x)=\frac{1}{nb_n}\sum^n_{j=1} K(\frac{x-X_j}{b_n}), }$ , where 0  <  b _n → 0 are bandwidths and K is some kernel function. In this paper, we study the uniformly Berry–Esseen bounds for these estimators of f(x). In particular, by choice of the bandwidths, the Berry–Esseen bounds of the estimators attain ${O\left((\log n/n)^{1/6}\right)}$ .