A refined Jensen's inequality in Hilbert spaces and empirical approximations
Let be a convex mapping and a Hilbert space. In this paper we prove the following refinement of Jensen's inequality: for every A,B such that and B[subset of]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 . 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.
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Volume (Year): 100 (2009)
Issue (Month): 5 (May)
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