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

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: E (f | X ∈ A) ≥ E (f | X ∈ B) 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 [S. Karlin, A. Novikoff, Generalized convex inequalities, Pacific J. Math. 13 (1963) 1251-1279], 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 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.