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Asymptotically Optimal Balloon Density Estimates

Author

Listed:
  • Hall, P.
  • Huber, C.
  • Owen, A.
  • Coventry, A.

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.

Suggested Citation

  • Hall, P. & Huber, C. & Owen, A. & Coventry, A., 1994. "Asymptotically Optimal Balloon Density Estimates," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 352-371, November.
  • Handle: RePEc:eee:jmvana:v:51:y:1994:i:2:p:352-371
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    Cited by:

    1. Cheng, Ming-Yen & Hall, Peter, 2003. "Reducing variance in nonparametric surface estimation," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 375-397, August.

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