On estimation of non-smooth functionals
- Lepskii, Oleg V.
- Nemirovski, Arkadi
- Spokoiny, Vladimir
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.
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