ABSTRACT
There is a vast literature on Berry-Esseen bounds for iid r.v.'s when the empirical d.f. is involved. Under PA, Wood (1983) obtained such a bound with maximum order of magnitude O(n-115) when the r.v.'s are indexed by integers, and Shao (1986) with order O(n-114 ) in a random field framework. Finally, Birkel (l986b) obtained a rate O(n-1/ 2 logn), under certain conditions, and showed that this rate cannot be improved. In Cai and Roussas (1995a), the empirical d.f. is replaced by the smooth kernel-type d.f. estimate F;,(x) referred to above, and, under the basic assumption ofPA or NA and additional suitable conditions, Berry-Esseen bounds with rates are obtained for vn[F11 (x)- O'F,(x)]. The rates obtained depend on the assumptions made and hinge heavily on the quantity u(r) = :L;::,,.Icov(X1, X1+1 1 11 . The reference Hall (1982) was quite helpful in some derivations.
