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Inference and risk measurement with the pari-mutuel model. (English) Zbl 1237.91136
Summary: We explore generalizations of the pari-mutuel model (PMM), a formalization of an intuitive way of assessing an upper probability from a precise one. We discuss a naive extension of the PMM considered in insurance, compare the PMM with a related model, the total variation model, and generalize the natural extension of the PMM introduced by P. Walley and other pertained formulae. The results are subsequently given a risk measurement interpretation: in particular it is shown that a known risk measure, tail value at risk (TVaR), is derived from the PMM, and a coherent risk measure more general than TVaR from its imprecise version. We analyze further the conditions for coherence of a related risk measure, conditional tail expectation. Conditioning with the PMM is investigated too, computing its natural extension, characterising its dilation and studying the weaker concept of imprecision increase.

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI
[1] Baroni, P.; Pelessoni, R.; Vicig, P., Generalizing Dutch risk measures through imprecise previsions, International journal of uncertainty, fuzziness and knowledge-based systems, 17, 2, 153-177, (2009) · Zbl 1162.91396
[2] de Cooman, G.; Troffaes, M.C.M.; Miranda, E., N-monotone lower previsions, Journal of intelligent & fuzzy systems, 16, 253-263, (2005)
[3] de Finetti, B., Theory of probability, vol. 1, (1974), Wiley
[4] Denneberg, D., Non-additive measure and integral, (1994), Kluwer · Zbl 0826.28002
[5] Denuit, M.; Dhaene, J.; Goovaerts, M.; Kaas, R., Actuarial theory for dependent risks: measures, orders and models, (2005), Wiley
[6] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling extremal events for insurance and finance, (1999), Springer-Verlag
[7] Gerber, H.U., An introduction to mathematical risk theory, (1979), Huebner Foundation · Zbl 0431.62066
[8] Herron, T.; Seidenfeld, T.; Wasserman, L., Divisive conditioning: further results on dilation, Philosophy of science, 64, 3, 411-444, (1996)
[9] Miranda, E.; de Cooman, G.; Quaeghebeur, E., Finitely additive extensions of distribution functions and moment sequences: the coherent lower prevision approach, International journal of approximate reasoning, 48, 1, 132-155, (2008) · Zbl 1204.60003
[10] Pelessoni, R.; Vicig, P., Imprecise previsions for risk measurement, International journal of uncertainty, fuzziness and knowledge-based systems, 11, 393-412, (2003) · Zbl 1074.91030
[11] Pelessoni, R.; Vicig, P., Williams coherence and beyond, International journal of approximate reasoning, 50, 4, 612-626, (2009) · Zbl 1214.68403
[12] Pelessoni, R.; Vicig, P., Bayes’ theorem bounds for convex lower previsions, Journal of statistical theory and practice, 3, 1, 85-101, (2009) · Zbl 1211.62012
[13] R. Pelessoni, P. Vicig, M. Zaffalon. The Pari-Mutuel Model, in: Proceedings of the ISIPTA ’09, Durham, UK, 2009, pp. 347-356. · Zbl 1237.91136
[14] Seidenfeld, T.; Wasserman, L., Dilation for sets of probabilities, The annals of statistics, 21, 3, 1139-1154, (1993) · Zbl 0796.62005
[15] P. Walley, Coherent Lower (and upper) Probabilities, Research Report, University of Warwick, Coventry, 1981.
[16] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall · Zbl 0732.62004
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