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

Let f:X→Rf:X→R be a convex mapping and XX a Hilbert space. In this paper we prove the following refinement of Jensen’s inequality: E(f|X∈A)≥E(f|X∈B)E(f|X∈A)≥E(f|X∈B) for every A,BA,B such that E(X|X∈A)=E(X|X∈B)E(X|X∈A)=E(X|X∈B) and B⊂AB⊂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=RX=R. The inverse implication is also true if PP 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 nn-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.