Skip to main content
SpringerLink
Log in

Symmetric coherent upper conditional prevision defined by the Choquet integral with respect to Hausdorff outer measure

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

In a metric space symmetric fuzzy measures defined on the class of all subsets are introduced. Coherent upper conditional probabilties defined by Hausdorff outer measures are symmetric and distorted coherent upper conditional probabilities defined by Hausdorff outer measures with concave distortion are proven to be symmetric. Null events and symmetric events with respect to coherent upper conditional probabilities defined by Hausdorff outer measures are characterized. Coherent upper conditional prevision defined as Choquet integral with respect to Hausdorff outer measure is symmetric because it is invariant with respect to equimeasurable random variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Billingsley, P. (1986). Probability and measure. London: Wiley.

    Google Scholar 

  • Choquet, G. (1953). Theory of capacities. Annales de l’Institut Fourier, 5, 131–295.

    Article  Google Scholar 

  • de Cooman, G., Quaeghebeur, E., & Miranda, E. (2009). Exchangeable lower previsions. Bernulli, pp. 721–735.

  • de Finetti, B. (1972). Probability, induction and statistics. New York: Wiley.

  • de Finetti, B. (1974). Theory of probability. London: Wiley.

  • Dellacherie, C. (1972). Ensembles analytiques, capacities, mesures de Hausdorff. In Lectures notes in mathematics. Berlin: Springer.

  • Denneberg, D. (1994). Non-additive measure and integral. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Doria, S. (2007). Probabilistic independence with respect to upper and lower conditional probabilities assigned by Hausdorff outer and inner measures. International Journal of Approximate Reasoning, 46, 617–635.

    Article  Google Scholar 

  • Doria, S. (2012). Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdorff outer measure. Annals of Operations Research, pp. 33–48.

  • Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. The Annals of Probability, 3, 89–99.

    Article  Google Scholar 

  • Falconer, K. J. (1986). The geometry of fractals sets. Cambridge: Cambridge University Press.

    Google Scholar 

  • Kadane, J. B., & Wasserman, L. (1996). Coherent choquet capacities. The Annals of Statistics Symmertric, 24(3), 1250–1264.

    Article  Google Scholar 

  • Marinacci, M. (1999). Upper probabilities and additivity, Sankhyā. The Indian Journal of Statistics, 61(Series A. Pt., 3), 358–361.

    Google Scholar 

  • Miranda, P., Grabish, M., & Gil, P. (2002). p-Symmetric fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10, 105–123.

    Article  Google Scholar 

  • Narukawa, Y., & Torra, V. (2005). Fuzzy measure and probability distributions: Distorted probabilities. IEEE Transactions on Fuzzy Systems, 13(5), 617–629.

    Article  Google Scholar 

  • Narukawa, Y., Torra, V., & Sugeno, M. (2012). Choquet integral with respect to a symmetric fuzzy measure of a function on the real line. Annals Operations Research. doi:10.1007/s10479-012-1166-6.

  • Pap, E. (2002). \(\sigma \)-null-additive set functions. Novi Sad Journal of Mathematics, 32(1), 47–57.

    Google Scholar 

  • Regazzini, E. (1985). Finitely additive conditional probabilities. Rendiconti del Seminario Matematico e Fisico di Milano, 55, 69–89.

    Article  Google Scholar 

  • Regazzini, E. (1987). De Finetti’s coherence and statistical inference. The Annals of Statistics, 15(2), 845–864.

    Article  Google Scholar 

  • Rogers, C. A. (1970). Hausdorff measures. Cambridge: Cambridge University Press.

    Google Scholar 

  • Torra, V. (1997). The weighted OWA operator. International Journal of Intelligent Systems, 12, 153–166.

    Article  Google Scholar 

  • Walley, P. (1981). Coherent lower (and upper) probabilities, statistics research report. University of Warwick.

  • Walley, P. (1991). Statistical reasoning with imprecise probabilities. London: Chapman and Hall.

    Book  Google Scholar 

  • Wasserman, L., & Kadane, J. B. (1992). Symmertric upper probabilities. The Annals of Statistics, 20(4), 1720–1736.

    Article  Google Scholar 

Download references

Acknowledgments

The author is grateful to two anonymous reviewers for their useful comments.

Author information

Authors and Affiliations

  1. Department of Engineering and Geology, University G. d’Annunzio, Chieti, Italy

    Serena Doria

Authors
  1. Serena Doria
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serena Doria.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Doria, S. Symmetric coherent upper conditional prevision defined by the Choquet integral with respect to Hausdorff outer measure. Ann Oper Res 229, 377–396 (2015). https://doi.org/10.1007/s10479-014-1752-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-014-1752-x

Keywords

Search

Navigation