On estimation of the Lr norm of a regression function

Abstract

Let a Hölder continuous function f be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, Lr norms ∥f∥r of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L2) 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.