A Refined Jensen's Inequality in Hilbert Spaces and Empirical Approximations

Abstract

Let f : X → R be a convex mapping and X a Hilbert space. In this paper we prove the following refinement of Jensen's inequality:for every A, B such that E(X |X ∈ A ) = E(X |X ∈ B ) and B ⊂ A. Expectations of Hilbert space valued random elements are defined by means of the Pettis integrals. Our result generalizes a result of Karlin and Novikov (1963), who derived it for X = 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 a 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.