Orderings of risks: A comparative study via stop-loss transforms. (English) Zbl 0855.62095

Summary: The relations between the following concepts for ordering risks are investigated: stochastic dominance, stop-loss order, convex order and being more dangerous. Using characterizations via stop-loss transforms, we give an elementary proof of the separation theorem for stop-loss order, and we correct a mistake in a result of E. A. van Heerwaarden [Ordering of risks: Theory and actuarial applications. Tinbergen Inst. Ser. 20 (1991)] on the connection between stop-loss order and being more dangerous. This is done by introducing a new notion of convergence for distributions. Moreover, we consider lattice properties of these orders.


62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
62E10 Characterization and structure theory of statistical distributions
Full Text: DOI


[1] Bühlmann, H.; Gagliardi, B.; Gerger, H.U.; Straub, E., Some inequalities for stop-loss premiums, ASTIN bulletin, 9, 75-83, (1977)
[2] De Groot, M.H., Optimal statistical decisions, (1970), MacGraw-Hill New York
[3] Elton, J.; Hill, T.P., Fusions of a probability distribution, Annals of probability, 20, 421-454, (1992) · Zbl 0747.60009
[4] Goovaerts, M.J.; Kaas, R.; van Heerwaarden, A.E.; Bauwelinckx, T., Effective actuarial methods, (1990), North Holland Amsterdam
[5] Heilmann, W.R.; Schröter, K.J., Orderings of risks and their actuarial applications, (), 157-173 · Zbl 0755.62077
[6] Kaas, R.; van Heerwaarden, A.E., Stop-loss order, unequal means, and more dangerous distributions, Insurance: mathematics and economics, 11, 71-77, (1992) · Zbl 0752.62073
[7] Karlins, S.; Novikoff, A., Generalized convex inequalities, Pacific journal of mathematics, 13, 1251-1279, (1963) · Zbl 0126.28102
[8] Kertz, R.P.; Rösler, U., Stochastic and convex orders and lattices of probability measures, Israel journal of mathematics, 77, 129-164, (1992) · Zbl 0771.60013
[9] Lehmann, E., Ordered families of distributions, Annals of mathematical statistics, 26, 399-419, (1955) · Zbl 0065.11906
[10] Makowski, A.M., On an elementary characterization of the increasing convex ordering, with an application, Journal of applied probability, 31, 834-840, (1994) · Zbl 0821.60024
[11] Ohlin, J., On a class of measures for dispersion with application to optimal insurance, ASTIN bulletin, 5, 249-266, (1969)
[12] Rockafellar, R.T., Convex analysis, (1970), Princeton Univ. Press Princeton, New Jersey · Zbl 0229.90020
[13] Rolski, T., Order relations in the set of probability distribution functions and their applications in queueing theory, Dissertationes mathematicae, 132, (1976) · Zbl 0357.60025
[14] Ross, S.M., Stochastic processes, (1983), Wiley New York · Zbl 0555.60002
[15] Rothschild, M.; Stiglitz, J.E., Increasing risk, I: A definition, Journal of economic theory, 2, 225-243, (1970)
[16] Shaked, M.; Shanthikumar, J.G., Stochastic orders and their applications, (1994), Academic Press London · Zbl 0806.62009
[17] Stoyan, D., Bounds for the extrema of the expected value of a convex function of independent random variables, Studia scientiarum mathematicarum hungarica, 8, 153-159, (1973) · Zbl 0274.60013
[18] Stoyan, D., Comparison methods for queues and other stochastic models, (1983), Wiley Chichester
[19] Strassen, V., The existence of probability measures with given marginals, Annals of mathematical statistics, 36, 423-439, (1965) · Zbl 0135.18701
[20] Sundt, B., An introduction to non-life insurance mathematics, (1991), Verlag Versicherungswirtschaft e.V Karlsruhe · Zbl 0727.62102
[21] van Heerwaarden, A.E., Ordering of risks: theory and actuarial applications, () · Zbl 0711.62095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.