On estimation of the Lr norm of a regression function

• Published in: Probability theory and related fields 1999-02, Vol.113 (2), p.221-253
• Main Authors: LEPSKI, O, NEMIROVSKI, A, SPOKOINY, V
• Format: Article
• Language: English
• Subjects: Convergence; Estimation; Exact sciences and technology; Integers; Linear inference, regression; Mathematics; Norms; Probability; Probability and statistics; Sciences and techniques of general use; Smoothness; Statistics
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s004409970006

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.