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A Weighted Bootstrap Procedure for Divergence Minimization Problems

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Analytical Methods in Statistics (AMISTAT 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 193))

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Abstract

Sanov-type results hold for some weighted versions of empirical measures, and the rates for those Large Deviation principles can be identified as divergences between measures, which in turn characterize the form of the weights. This correspondence is considered within the range of the Cressie–Read family of statistical divergences, which covers most of the usual statistical criterions. We propose a weighted bootstrap procedure in order to estimate these rates. To any such rate we produce an explicit procedure which defines the weights, therefore replacing a variational problem in the space of measures by a simple Monte Carlo procedure.

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Notes

  1. 1.

    We say a function is proper if its domain is non void.

  2. 2.

    The closedness of \(\varphi \) means that if \(a_{\varphi }\) or \(b_{\varphi }\) are finite numbers then \(\varphi (x)\) tends to \(\varphi (a_{\varphi })\) or \(\varphi (b_{\varphi })\) when \(x\downarrow a_{\varphi }\) or \(x\uparrow b_{\varphi }\), respectively.

  3. 3.

    We say a function \(\varphi \) is lower semi-continuous if the level sets \(\left\{ x\text { such that } \varphi (x)\le \alpha \right\} \), \(\alpha \in \mathbb {R}\) are closed.

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Authors and Affiliations

  1. Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie, Boîte Courrier 158, 4 place Jussieu, 75252, Paris Cedex 05, France

    Michel Broniatowski

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  1. Michel Broniatowski
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Correspondence to Michel Broniatowski .

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Editors and Affiliations

  1. Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic

    Jaromír Antoch

  2. Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic

    Jana Jurečková

  3. Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic

    Matúš Maciak

  4. Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic

    Michal Pešta

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Broniatowski, M. (2017). A Weighted Bootstrap Procedure for Divergence Minimization Problems. In: Antoch, J., Jurečková, J., Maciak, M., Pešta, M. (eds) Analytical Methods in Statistics. AMISTAT 2015. Springer Proceedings in Mathematics & Statistics, vol 193. Springer, Cham. https://doi.org/10.1007/978-3-319-51313-3_1

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