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

Abstract: Let be a convex mapping and a Hilbert space. In this paper we prove the following refinement of Jensen’s inequality: for every such that and . Expectations 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 . The inverse implication is also true if 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 -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. [Copyright &y& Elsevier]