Data di Pubblicazione:
2009
Citazione:
A refined Jensen's inequality in Hilbert spaces and empirical approximations / S. Leorato. - In: JOURNAL OF MULTIVARIATE ANALYSIS. - ISSN 0047-259X. - 100:5(2009), pp. 1044-1060.
Abstract:
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.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
60E15; 62G08; Convex regression function; Empirical measure; Jensen's inequality; Linearly ordered classes of sets; Pettis integral; Supporting hyperplane
Elenco autori:
S. Leorato
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