Mathematics Subject Classiication. 62G077 Secondary 62G20
Let a function f be observed with noise. In the present paper we concern the problem of nonparametric estimation of some non-smooth functionals of f , more precisely, L r-norm kfk r of f. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: estimation of a smooth (diierentiable in L 2) functional or estimation of a singular functional like the value of f at a certain point or the maximum of f. In the rst case, the rate of estimation is typically n
... ion is typically n ;1=2 , n being the numb e r o f o b s e r v ations. In the second case, the rate of functional estimation coincides with the nonparametric rate of estimation of the whole function f in the corresponding norm. We s h o w that the case of estimation of kfk 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. F or r even integer, the rate occurs to be n ;=(2+1;1=r) where is the degree of smoothness. If r is not even integer, then the nonparametric rate n ;=(2+1) can be improved only by some logarithmic factor.