WIAS Preprint No. 297, (1996)

On estimation of non-smooth functionals


  • Lepskii, Oleg V.
  • Nemirovski, Arkadi
  • Spokoiny, Vladimir
    ORCID: 0000-0002-2040-3427

2010 Mathematics Subject Classification

  • 62G07 62G20


  • non-smooth functionals, integral norm, rate of estimation




Let a function ƒ be observed with noise. In the present paper we concern the problem of nonparametric estimation of some non-smooth functionals of ƒ, more precisely, Lr -norm ∥ƒ∥r of ƒ. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: estimation of a smooth (differentiable in L2) functional or estimation of a singular functional like the value of ƒ at a certain point or the maximum of ƒ. In the first case, the rate of estimation is typically n-1/2 , n being the number of observations. In the second case, the rate of functional estimation coincides with the nonparametric rate of estimation of the whole function ƒ in the corresponding norm. We show that the case of estimation of ∥ƒ∥r is in some sense intermediate between the above extreme two. The optimal rate of estimation is worse than n-1/2 but better than the usual nonparametric rate. 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 even integer, then the nonparametric rate n -β/(2β+1) can be improved only by some logarithmic factor.

Appeared in

  • Probability Theory and Related Fields, 113 (1999), 221-253. under the title: On estimation of the Lr-norm of a regression functionals.

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