MASS SHIFTING ROLES OF NEGATIVE KERNEL MASS IN DENSITY ESTIMATION

MICHAEL STURGEON

Mathematics Department, Salem State College, 45 Blaisdell Drive, Carlisle, MA 01741, U.S.A.

(Received March 13, 1995; revised October 9, 1995)

Abstract. f(x) is a univariate density in C^{4} with bounded support. For any n and sufficiently small kernel bandwidths, the symmetric appendage of any negative mass, −U, to any smooth unimodal symmetric kernel of order p = 2 shifts expected estimator mass from regions where f′′(x) > 0 to regions where f′′(x) < 0. For large n, the mean automatic kernel adaptation induced by −U is analyzed in the simplest MISE reduction scenario: The symmetric appendage of −U to the uniform kernel K(x, X) over MISE-optimal bandwidths reduces MISE by shifting K(x, X) mass asymmetrically across the observation X in the direction of decreasing |f′′(x)|.

Key words and phrases: Kernel adaptation, higher order kernels, induced kernel asymmetry.