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Some Models of Inference in the Risk Theory from a Bayesian Viewpoint

Published online by Cambridge University Press:  29 August 2014

Luciano Daboni
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Our purpose is to introduce some models of inference for risk processes. The bayesian viewpoint is adopted and for our treatment the concepts of exchangeability and partial exchangeability (due to B. de Finetti, [6], [7]) are essential.

We recall the definitions:

The random variables of a sequence (X1, X2 …) are exchangeable if, for every n, the joint distribution of n r.v. of the sequence is always the same, whatever the n r.v. are and however they are permuted.

From a structural point of view an exchangeable process X1, X2 … can be intended as a sequence of r.v. equally distributed among which a “stochastic dependence due to uncertainty” exists. More precisely the Xi are independent conditionally on any of a given set (finite or not) of exhaustive and exclusive hypothesis. These hypotheses may concern, for instance, the values of a parameter (number or vector) on which the common distribution, of known functional form, of Xi depends. We shall restrict ourselves to this case. Therefore, we shall assume that, conditionally on each possible value θ of a parameter Θ, the Xi are independent with F(x/θ) as known distribution function. According to the bayesian approach, a probability distribution on Θ must be assigned.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 8 , Issue 1 , September 1974 , pp. 38 - 56
Copyright
Copyright © International Actuarial Association 1974

References

[1]Bühlmann, H., Optimale Prämienstufensysteme, Mitteilung der Vereinigung schweizerischer Versicherungsmathematiker, Bd 64, 1964.Google Scholar
[2]Bühlmann, H., Kollektive Risikotheorien, Mitteilung der Vereinigung schweizerischer Versicherungsmathematiker, Bd 67, 1967.Google Scholar
[3]Buhlmann, H., Austauschbare stochatische Variabeln und ihre Grenzwertsaetze, 1960 (Univ of California Press)Google Scholar
[4]Crisma, LUna propietà caratteristica dei processi di Pólya, 1970 (Studie ricerche, VII, Univ Parma)Google Scholar
[5]Daboni, L, Processi markoffiani di puro ingresso con intervalli di attesa scambiabili, 1969 (Studi e ricerche, VI, Univ Parma)Google Scholar
[6]de FiNetti, B, La prevision ses lois logiques, ses sources subjectives, 1937 (An Inst H Poincaré)Google Scholar
[7]de FiNetti, B, Teoria delle probabilità, 1970, Einaudi TorinoGoogle Scholar
[8]Lindley, D V., Bayesian statistics A review Published by Siam Philadelphia, 1971Google Scholar
[9]Lindley, D V, Introduction to Probability and Statistics from a Bayesian Viewpoint Pt I Probability Pt 2 Inference 1965 Univ Press CambridgeGoogle Scholar
[10]LundbΓrg, O, On random processes and their application to sickness and accident statistics, 1940 Univ of Stockholm (2nd edit Uppsala 1964)Google Scholar
[11]Seal, H, Stochastic theory of a Risk Business, 1969 J WileyGoogle Scholar
[12]Strudthoff, M, Un processo markoffiano di puro ingresso a parametro continuo ottenuto come estensione del processo discreto della frequenza di successo nel modello ipergeometrico 1973 Ist Mat Univ ⅂rieste Vol VGoogle Scholar
[13]Wedlin, A, Osservazioni su alcuni processi inferenziali in ipotesi di scambiabilità parziale Università di Trieste, 1972Google Scholar
[14]Zellner, A, An introduction to Bayesian Inference in Econometrics, 1971, J Wiley N YGoogle Scholar