A refined Jensen's inequality in Hilbert spaces and empirical approximations

Let f : (sic) -> R be a convex mapping and (sic) a Hilbert space. In this paper we prove the following refinement of Jensen's inequality: E(f vertical bar X is an element of A) >= E(f vertical bar X is an element of B) for every A, B such that E(f vertical bar X is an element of A) >= E(f vertical bar X is an element of B) and B subset of A. Expectation 5 of Hilbert-space-valued random elements are defined by means of the Pettis integrals. Our result generalizes a result of [S. Karlin, A. Novikoff, Generalized convex inequalities, Pacific J. Math. 13 (1963) 1251-1279], who derived it for (sic) = R. The inverse implication is also true if P is an absolutely continuous probability measure. A convexity criterion based on the Jensen-type inequalities follows and we study its asymptotic accuracy when the empirical distribution function based on an n-dimensional sample approximates the unknown distribution function. Some statistical applications are addressed, such as nonparametric estimation and testing for convex regression functions or other functionals. (C) 2008 Elsevier Inc. All rights reserved.