Mass shifting roles of negative kernel mass in density estimation

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)|.