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On estimation of the L r norm of a regression function

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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.

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Received: 6 February 1996hinspaceairsp/Revised version: 10 June 1998

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Lepski, O., Nemirovski, A. & Spokoiny, V. On estimation of the L r norm of a regression function. Probab Theory Relat Fields 113, 221–253 (1999).

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  • Mathematics Subject Classification (1991): 62G07; Secondary 62G20
  • Key words: Non-smooth functionals – Integral norm – Rate of estimation