Asymptotically Optimal Balloon Density Estimates

P. Hall, C. Huber, A. Owen and A. Coventry

Journal of Multivariate Analysis, 1994, vol. 51, issue 2, 352-371

Abstract: Given a sample of n observations from a density [latin small letter f with hook] on d, a natural estimator of [latin small letter f with hook](x) is formed by counting the number of points in some region surrounding x and dividing this count by the d dimensional volume of . This paper presents an asymptotically optimal choice for . 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 Lp balls for even positive integers p.

Date: 1994
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