A refined Jensen's inequality in Hubert 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 ... Expectations of Hilbert-space-valued random elements are defined by means of the Pettis integrals. Our result generalizes a result...

Full description

Saved in:
Bibliographic Details
Published inJournal of multivariate analysis Vol. 100; no. 5; p. 1044
Main Author Leorato, S
Format Journal Article
LanguageEnglish
Published New York Taylor & Francis LLC 01.05.2009
Subjects
Approximation
Asymptotic methods
Calculus
Mathematical functions
Probability
Studies
Vector space
Online AccessGet full text

Cover

More Information
Summary: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 ... 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. (ProQuest: ... denotes formulae/symbols omitted.)
ISSN:0047-259X
1095-7243