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A refined Jensen's inequality in Hilbert spaces and empirical approximations

  • Leorato, S.

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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Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 100 (2009)
Issue (Month): 5 (May)
Pages: 1044-1060

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Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:1044-1060
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  1. 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.
  2. 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.
  3. 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.
  4. Rodríguez Poo, Juan M. & Ferreira García, María Eva & Orbe Mandaluniz, Susan, 2001. "Nonparametric estimation of time varying parameters under shape restrictions," BILTOKI 2001-02, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
  5. M. Menéndez & D. Morales & L. Pardo & I. Vajda, 2001. "Minimum Divergence Estimators Based on Grouped Data," Annals of the Institute of Statistical Mathematics, Springer, vol. 53(2), pages 277-288, June.
  6. 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.
  7. 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.
  8. 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.
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