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On estimation of the L r norm of a regression function
Authors:O Lepski  A Nemirovski  V Spokoiny
Institution:(1) Humboldt University, SFB 373, Spandauer Str. 1, D-10178 Berlin Germany, DE;(2) Technion–Israel Institute of Technology, Haifa 32000, Israel, IL;(3) Weierstrass Institute, Mohrenstr. 39, D-10117 Berlin, Germany. e-mail:spokoiny@wias-berlin.de, DE
Abstract:f be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, L r norms ||f|| r of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L 2 ) functional or estimating a singular functional like the value of f at certain point or the maximum of f. In the first case, the convergence rate typically is n −1/2, n being the number of observations. In the second case, the rate of convergence coincides with the one of estimating the function f itself in the corresponding norm. We show that the case of estimating ||f|| r is in some sense intermediate between the above extremes. The optimal rate of convergence is worse than n −1/2 but is better than the rate of convergence of nonparametric estimates of f. The results depend on the value of r. For r even integer, the rate occurs to be n −β/(2β+1−1/r) where β is the degree of smoothness. If r is not an even integer, then the nonparametric rate n −β/(2β+1) can be improved, but only by a logarithmic in n factor. Received: 6 February 1996hinspaceairsp/Revised version: 10 June 1998
Keywords:Mathematics Subject Classification (1991): 62G07  Secondary 62G20
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