Berry-Esseen bounds for density estimates under NA assumption

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 (f) over cap (n)(x) = 1/n root b(n) (n)Sigma(j=1) b(j)(-1/2) K(x-X-j/b(j)), (f) over tilde (n)(x) = 1/n = 1/n (n)Sigma(j=1) 1/b(j)K(x-X-j/b(j)) and the Rosenblatt-Parzen's kernel estimator of f(x) is defined by f(n)(x) = 1/nb(n) Sigma(n)(j=1) K(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 ((log n/n)(1/6)).