General Bootstrap for Dual ？-Divergence Estimates.pdf.pdf

Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2012, Article ID 834107, 33 pages doi: Research Article General Bootstrap for Dual φ-Divergence Estimates Salim Bouzebda 1, 2 and Mohamed Cher? 2 1Laboratoire de Math ′ematiques Appliqu ′ees, Universit ′e de Technologie pi `egne, . 529, pi `egne Cedex, France 2LSTA, Universit ′e Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France Correspondence should be addressed to Salim Bouzebda, salim.******@ Received 30 May 2011; Revised 29 September 2011; Accepted 16 October 2011 Academic Editor: Rongling Wu Copyright q2012 S. Bouzebda and M. Cher?. This is an open access article distributed under the mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A general notion of bootstrapped φ-divergence estimates constructed by exchangeably weighting sample is introduced. Asymptotic properties of these generalized bootstrapped φ-divergence estimates are obtained, by means of the empirical process theory, which are applied to construct the bootstrap con?dence set with asymptotically correct coverage probability. Some of practical problems are discussed, including, in particular, the choice of escort parameter, and several examples of divergences are investigated. Simulation results are provided to illustrate the ?nite sample performance of the proposed estimators. 1. Introduction The φ-divergence modeling has proved to be a ?exible tool and provided a powerful statistical modeling framework in a variety of applied and theoretical contexts ?refer to ? 1 – 4 ? and the references therein ?. For good recent sources of references to the research literature in this area along with statistical applications, consult ? 2 , 5 ?. Unfortunately, in general, the limiting distribution of the estimators, or their functionals, based on φ-divergences depends crucially on the unkno