Ordering of distributions and risk measurement. (English) Zbl 0662.62110
The present paper is an attempt to bring together two branches of risk theory, viz. ordering of risks and premium calculation, and a topical section of decision theory, viz. risk measurement. We first introduce and compare several orderings of distributions. Then a recent model of risk measurement is introduced and shown to be related to two quantities - a probability of “loss” and a loss distribution - in the theory of risk processes. In the Poisson case exactly by this type of distribution one of the above developed order relations is generated. Finally we show how these results can be used to create principles of premium calculation.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
Keywords:
uniform distribution; exponential distribution; Pareto-distribution; stochastic dominance; risk theory; ordering of risks; decision theory; risk measurement; orderings of distributions; loss distribution; risk processes; Poisson case; premium calculationReferences:
| [1] | Bowers, N. L., Gerber, H. U., Hickman, J. C.,Jones, D. A. andNesbitt, C. J.: Risk Theory (Part 5 A Study Note). Society of Actuaries, 1982. |
| [2] | Fishburn P. C.: Foundations of Risk Measurement. I: Risk as Probable Loss. Man. Science 30, 1984, 396–406. · Zbl 0552.90004 · doi:10.1287/mnsc.30.4.396 |
| [3] | Heilmann, W.-R.: Transformations of Claim Distributions. Mitt. Verein. Schweiz. Vers.Math., 1985, 57–69. · Zbl 0585.62176 |
| [4] | Taillie, Ch.: Lorenz Ordering with the Generalized Gamma Family of Income Distributions. In: Th. Taillie et al. (eds.): Statistical Distributions in the Scientific Work. D. Reidel, 1981, 181–192. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
