KERNEL DENSITY ESTIMATION FOR LINEAR PROCESSES: ASYMPTOTIC NORMALITY AND OPTIMAL BANDWIDTH DERIVATION

MARC HALLIN^{1} AND LANH TAT TRAN^{2}

1Institut de Statistique, Université Libre de Bruxelles, Campus de la Plaine CP 210,

Boulevard du Triomphe, B-1050 Bruxelles, Belgium

2Department of Mathematics, College of Arts and Sciences, Indiana University, Rawles Hall, Bloomington, IN 47405-5701, U.S.A.

(Received January 4, 1994; revised September 21, 1995)

Abstract. The problem of estimating the marginal density of a linear process by kernel methods is considered. Under general conditions, kernel density estimators are shown to be asymptotically normal. Their limiting covariance matrix is computed. We also find the optimal bandwidth in the sense that it asymptotically minimizes the mean square error of the estimators. The assumptions involved are easily verifiable.

Key words and phrases: Density estimation, linear process, kernel, bandwidth, mean square error.