Limit of the quadratic risk in density estimation using linear methods

Kerkyacharian Gérard and Picard Dominique

Statistics & Probability Letters, 1997, vol. 31, issue 4, 299-312

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.

Keywords: Minimax; estimation; Mean; integrated; squared; error; density; estimation; Wavelet; orthonormal; bases; Besov; spaces (search for similar items in EconPapers)
Date: 1997
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