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Asymptotically Optimal Balloon Density Estimates

Authors
Journal
Journal of Multivariate Analysis
0047-259X
Publisher
Elsevier
Publication Date
Volume
51
Issue
2
Identifiers
DOI: 10.1006/jmva.1994.1067

Abstract

Abstract Given a sample of n observations from a density ƒ on R d , a natural estimator of ƒ( x) is formed by counting the number of points in some region R surrounding x and dividing this count by the d dimensional volume of R . This paper presents an asymptotically optimal choice for R . The optimal shape turns out to be an ellipsoid, with shape depending on x. An extension of the idea that uses a kernel function to put greater weight on points nearer x is given. Among nonnegative kernels, the familiar Bartlett-Epanechnikov kernel used with an ellipsoidal region is optimal. When using higher order kernels, the optimal region shapes are related to L p balls for even positive integers p.

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