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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|Date of creation:||24 Nov 2008|
|Date of revision:||24 Nov 2008|
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- Perlman, Michael D., 1974. "Jensen's inequality for a convex vector-valued function on an infinite-dimensional space," Journal of Multivariate Analysis, Elsevier, vol. 4(1), pages 52-65, March.
- Hall, Peter & Yatchew, Adonis, 2005. "Unified approach to testing functional hypotheses in semiparametric contexts," Journal of Econometrics, Elsevier, vol. 127(2), pages 225-252, August.
- Orbe, Susan & Ferreira, Eva & Rodriguez-Poo, Juan, 2005. "Nonparametric estimation of time varying parameters under shape restrictions," Journal of Econometrics, Elsevier, vol. 126(1), pages 53-77, May.
- Menendez, M. & Morales, D. & Pardo, L. & Vajda, I., 1995. "Divergence-Based Estimation and Testing of Statistical Models of Classification," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 329-354, August.
- M. Menéndez & D. Morales & L. Pardo & I. Vajda, 2001. "Minimum Divergence Estimators Based on Grouped Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(2), pages 277-288, June.
- Kozek, A. & Suchanecki, Z., 1980. "Multifunctions of faces for conditional expectations of selectors and Jensen's inequality," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 579-598, December.
- Jason Abrevaya & Wei Jiang, 2005. "A Nonparametric Approach to Measuring and Testing Curvature," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 1-19, January.
- Melanie Birke & Holger Dette, 2007. "Estimating a Convex Function in Nonparametric Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(2), pages 384-404.
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