ON SMOOTHED PROBABILITY DENSITY ESTIMATION

The main object of this paper is to study properties of the estimator (x)1n 1 K Cx—X~ fn=n ~ai a under the assumption of stationarity of the sequence (X7). 0. Introduction Suppose a sample of observations X1, X2, , Xn is identically distributed with density function f. Much research in recent years is concentrated on studying pro perties of the kernel estimator _ 1 n((x—X) fn(ti)—na n iK\an2),(0.1 where { a n } , n=1, 2, is a given sequence of positive numbers such that a n,--*0 (n—>co) and K is a given kernel. Recently properties of the estimator 77, are studied under the assumption of stationarity of the sample. See Masry [10] for the case of a sta tionary continuous-time process and Castellana and Leadbetter [4] for an approach using 3-sequences. In case of dependence it can be expected that properties of the estimator fn can be improved if the window width is not necessarily the same for each observa tion, that means the estimator fn(x)= 1 1----K(~)(0.2) n i=i a, ai is considered. Earlier research concerning fn in case of independent observations is done by Devroye [7], Samanta and Mugisha [11], who extended results of Yamato [13], and Davies [5]. One of the results (Theorem 1.3) in this paper is that (in case of dependence) for a suitable choice of the sequence { a 7,} the variance of the estimator fn is smaller than the variance of the usual estimator In. Moreover, under suitable assumptions, the estimator in is asymptotically normal. This is shown in Theorem 2.4 which contains a result about an estimator for f' as well. Finally Section 3 contains uniform con vergence results, weakly as well as strongly. * University of Petroleum & Minerals , Dhahran, Saudi Arabia.