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Limit of the quadratic risk in density estimation using linear methods

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  • Gérard, Kerkyacharian
  • Dominique, Picard

Abstract

We prove here that when estimating a density, using a kernel or a linear wavelet estimate, one can choose the smoothing parameter such that the limit when n tends to infinity of ?????? may be arbitrarily small for every density having a square integrable derivative. This choice consists in starting from the usual rate n-1/3 and then operate an oversmoothing proportional to the limit of the risk we want to obtain. Looking at the limit of the risk is another way of looking at the performances of estimators: We introduce here the maximal functional space where the results still stand. We show that this space contains the Sobolev spaces for instance. We also give a comparison with the standard minimax theory.

Suggested Citation

  • Gérard, Kerkyacharian & Dominique, Picard, 1997. "Limit of the quadratic risk in density estimation using linear methods," Statistics & Probability Letters, Elsevier, vol. 31(4), pages 299-312, February.
    • Handle: RePEc:eee:stapro:v:31:y:1997:i:4:p:299-312
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    References listed on IDEAS

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    1. Masry, Elias, 1994. "Probability density estimation from dependent observations using wavelets orthonormal bases," Statistics & Probability Letters, Elsevier, vol. 21(3), pages 181-194, October.
    2. Kerkyacharian, G. & Picard, D., 1992. "Density estimation in Besov spaces," Statistics & Probability Letters, Elsevier, vol. 13(1), pages 15-24, January.
    3. Kerkyacharian, Gérard & Picard, Dominique, 1993. "Density estimation by kernel and wavelets methods: Optimality of Besov spaces," Statistics & Probability Letters, Elsevier, vol. 18(4), pages 327-336, November.
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