LOCAL POLYNOMIAL FITTING WITH LONG-MEMORY, SHORT-MEMORY AND ANTIPERSISTENT ERRORS

JAN BERAN AND YUANHUA FENG

Department of Mathematics and Statistics, University of Konstanz, Universitätsstr. 10, Postfach 5560, D-78457 Konstanz, Germany, e-mail: jan.beran@uni-konstanz.de; yuanhua.feng@uni-konstanz.de

(Received May 12, 2000; revised October 30, 2000)

Abstract. Local polynomial smoothing for the trend function and its derivatives in nonparametric regression with long-memory, short-memory and antipersistent errors is considered. We show that in the case of antipersistence, the convergence rate of a nonparametric regression estimator is faster than for uncorrelated or short-range dependent errors. Moreover, it is shown that unified asymptotic formulas for the optimal bandwidth and the MSE hold for all of the three dependence structures. Also, results on estimation at the boundary are included. A bandwidth selector for nonparametric regression with different types of dependent errors is proposed. Its asymptotic property is investigated. The practical performance of the proposal is illustrated by simulated and real data examples.

Key words and phrases: Antipersistence, long-range dependence, local polynomial fitting, nonparametric regression, bandwidth selection.