A refined Jensen's inequality in Hilbert spaces and empirical approximations
نویسنده
چکیده
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 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.
منابع مشابه
Some Reverses of the Jensen Inequality for Functions of Selfadjoint Operators in Hilbert Spaces
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عنوان ژورنال:
- J. Multivariate Analysis
دوره 100 شماره
صفحات -
تاریخ انتشار 2009