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Asymptotic Limit of the Bayes Actions Set Derived from a Class of Loss Functions

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  • Abraham, Christophe

Abstract

Similarly to the determination of a prior in Bayesian Decision theory, an arbitrarily precise determination of the loss function is unrealistic. Thus, analogously to global robustness with respect to the prior, one can consider a set of loss functions to describe the imprecise preferences of the decision maker. In this paper, we investigate the asymptotic behavior of the Bayes actions set derived from a class of loss functions. When the collection of additional observations induces a decrease in the range of the Bayes actions, robustness is improved. We give sufficient conditions for the convergence of the Bayes actions set with respect to the Hausdorff metric and we also give the limit set. Finally, we show that these conditions are satisfied when the set of decisions and the set of states of nature are subsets of p.

Suggested Citation

  • Abraham, Christophe, 2001. "Asymptotic Limit of the Bayes Actions Set Derived from a Class of Loss Functions," Journal of Multivariate Analysis, Elsevier, vol. 79(2), pages 251-274, November.
    • Handle: RePEc:eee:jmvana:v:79:y:2001:i:2:p:251-274
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    References listed on IDEAS

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    1. Rubin Herman, 1987. "A Weak System Of Axioms For "Rational" Behavior And The Nonseparability Of Utility From Prior," Statistics & Risk Modeling, De Gruyter, vol. 5(1-2), pages 47-58, February.
    2. Christophe Abraham & Jean-Pierre Daurès, 1999. "Analytic approximation of the interval of Bayes actions derived from a class of loss functions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(1), pages 129-145, June.
    3. Strasser, Helmut, 1973. "On Bayes estimates," Journal of Multivariate Analysis, Elsevier, vol. 3(3), pages 293-310, September.
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    Cited by:

    1. Abraham, Christophe, 2005. "Asymptotics in Bayesian decision theory with applications to global robustness," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 50-65, July.

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