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MASS SHIFTING ROLES OF NEGATIVE KERNEL MASS

IN DENSITY ESTIMATION

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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.

**Source**
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