On estimation of non-smooth functionals

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 parallel f parallel _r of f. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: Estimation of a smooth (differentiable 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 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 f in the corresponding norm. We show that the case of estimation of parallel f parallel _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"-"#beta#"/"("2"#beta#"+"1"-"1"/"r") where #beta# is the degree of smoothness. If r is not even integer, then the nonparametric rate n"-"#beta#"/"("2"#beta#"+"1") can be improved only by some logarithmic factor. (orig.)Available from TIB Hannover: RR 5549(297)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

On estimation of non-smooth functionals

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 parallel f parallel _r of f. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: Estimation of a smooth (differentiable 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 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 f in the corresponding norm. We show that the case of estimation of parallel f parallel _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"-"#beta#"/"("2"#beta#"+"1"-"1"/"r") where #beta# is the degree of smoothness. If r is not even integer, then the nonparametric rate n"-"#beta#"/"("2"#beta#"+"1") can be improved only by some logarithmic factor. (orig.)Available from TIB Hannover: RR 5549(297)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman