Blätter der DGVFM

, Volume 24, Issue 3, pp 449–463

# Higher degree stop-loss transforms and stochastic orders — (I) Theory

• Werner Hürlimann
Article

## Summary

The higher degree stop-loss transforms and their logarithmic derivatives, called higher degree stop-loss rate functions, are studied to get insight into the hierarchical theory of the higher degree stop-loss orders and related stochastic orders. Based on differential-integral recursive relationships, we derive in a simple way two characterization results by Gupta and Gupta (1983), which state that the higher degree stop-loss transforms and the higher degree stop-loss rate functions uniquely determine a distribution function. Classes ISLR (n) of distributions with an increasing stop-loss rate function of degree n are considered, and it is shown that ISLR (n) implies ISLR (n+1). This result generalizes the well-known fact by Bryson and Siddiqui (1969) that a distribution with an increasing failure rate has necessarily a decreasing mean residual life. Necessary and sufficient conditions, which guarantee that ISLR (n+1) implies ISLR (n), are formulated. Using notions of higher degree stop-loss rate order and higher degree stop-loss rate dangerousness order, sufficient conditions for a higher degree stop-loss order relation are established. Two new sign change characterizations of the higher degree stop-loss order by means of higher degree stop-loss transforms and higher degree stop-loss rate functions are derived. Applications in actuarial mathematics follow in part (II) of the present work.

## Keywords

Hazard Rate Stochastic Dominance Residual Life Stochastic Order Hazard Rate Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

# Stop-Loss Transformierte eines höheren Grades und stochastische Ordnungen - (I) Theorie

## Zusammenfassung

Die Stop-Loss Transformierten eines höheren Grades und ihre logarithmische Ableitungen, genaant Stop-Loss Raten eines höheren Grades, werden untersucht, um Einsicht in die Hierarchie der Stop-Loss Ordnungen und verwandte stochastische Ordnungen zu erlangen. Mit Hilfe von Differential und Integral rekursive Relationen werden zwei Charakterisierungen von Gupta und Gupta (1983) auf einfache Weise hergeleitet. Diese Resultate zeigen, daß eine Verteilungsfunktion eindeutig durch eine Stop-Loss Transformierte oder Stop-Loss Rate eines höheren Grades definiert ist. Klassen ISLR (n) von Verteilungen mit einer wachsenden Stop-Loss Rate des Grades n werden betrachtet. Es wird gezeigt, daß die Eigenschaft ISLR (n) die Eigenschaft ISLR (n+1) zur Folge hat, was das wohlbekannte Resultat von Bryson and Siddiqui (1969) für den Fall n=0 verallgemeinert. Hinreichende Bedingungen für eine Stop-Loss Ordnung eines höheren Grades werden anhand von Stop-Loss Raten Ordnungen und Stop-Loss Raten Gefährlichkeitsordnungen formuliert. Zwei neue Charakterisierungen der Stop-Loss Ordnungen eines höheren Grades, welche die Vorzeichenänderungen der Stop-Loss Transformierten und der Stop-Loss Raten berücksichtigt, werden aufgestellt. Anwendungen in Versicherungsmathematik folgen in Teil (II) dieser Abhandlung.

## References

1. Ali, M. M. (1974): Stochastic ordering and kurtosis measure. Journal of the American Statistical Association 69, 543–545.
2. Bain, L. J. (1978): Statistical Analysis of Reliability and Life-Testing Models. Marcel Dekker.Google Scholar
3. Barlow, R. E. andProschan, F. (1965): The Mathematical Theory of Reliability. J. Wiley. Reprinted (1996). Classics in Applied Mathematics 17. Society for Industrial and Applied Mathematics.Google Scholar
4. Barlow, R. E. andProschan, F. (1975): Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.
5. Bhattacharjee, M. C. (1982): The class of mean residual lives and some consequences. SIAM Journal of Algebraic and Discrete Methods 3(1), 56–65.
6. Block, H. W., Savits, Th. H., andSingh, H. (1998): The reversed hazard rate function. Probability in the Engineering and Informational Sciences, 69–90.Google Scholar
7. Bryson, M. C. andSiddiqui, M. M. (1969): Some criteria for aging. Journal of the American Statistical Association 64, 1472–1483.
8. Bühlmann, H., Gagliardi, B., Gerber, H. U., andStraub, E. (1977): Some inequalities for stop-loss premiums. ASTIN Bulletin 9, 75–83.Google Scholar
9. Denuit, M., Lefèvre, Cl., andShaked, M. (1997): The s-convex orders among real random variables, with applications. Preprint de l’Institut de Statistique et de Recherche Opérationelle, Série Probabilités et Statistiques 65, Mai 1997.Google Scholar
10. Denuit, M., Lefèvre, Cl., andDe Vylder, F. (1998): Extremal generators and extremal distributions for the continuous s-convex stochastic orderings. Preprint de l’Institut de Statistique et de Recherche Opérationelle, Série Probabilités et Statistiques 90, Mars 1998.Google Scholar
11. Eckhoudt, L. andGollier, Ch. (1995): Demand for risky assets and the monotone probability ratio. Journal of Risk and Uncertainty 11, 113–122.
12. Fishburn, P. C. (1980): Stochastic dominance and moments of distributions. Mathematics of Operations Research 5, 94–100.
13. Fishburn, P. C. andPorter, B. (1976): Optimal portfolios with one safe and one risky asset: effects of changes in rate of return and risk. Management Science 22, 1069–1073.
14. Fishburn, P. C. andVickson, R. G. (1978): Theoretical foundations of stochastic dominance. In Whitmore, G. A. and Findlay, M. C. (Eds.). Stochastic dominance. D. C. Health and Co., Lexington (Massachusetts), 37–113.Google Scholar
15. Gerber, H. U. (1979): An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8. R.D. Irwin, Homewood, Illinois.
16. Goovaerts, M. J., De Vylder, F., andHaezendonck, J. (1984): Insurance Premiums. North-Holland.Google Scholar
17. Gupta, R. C. (1975): On characterizations of distributions by conditional expectations. Communications in Statistics 4(1), 99–103.Google Scholar
18. Gupta, P. L. andGupta, R. C. (1983): On the moments of residual life in reliability and some characterization results. Communications in Statistics — Theory and Methods 12(4), 449–461.
19. Hall, W. J. andWellner, J. A. (1981): Mean residual life. In Csörgö, M., Dawson, D. A., Rao, J. N. K., Saleh, A. K. Md. E. (Eds.). Statistics and Related Topics. North-Holland, 169–184.Google Scholar
20. Heerwaarden, van A. E. (1991): Ordering of risks: theory and actuarial applications. Ph.D. Thesis, Tinbergen Research Series no. 20, Amsterdam.Google Scholar
21. Heilmann, W.-R. (1984): Charakterisierungen von Lebensdauer- und Schadenhöhenverteilungen. Blätter der Deutschen Gesellschaft für Versicherungsmathematik, 409–422.Google Scholar
22. Hesselager, O. (1996): A unification of some order relations. Insurance: Mathematics and Economics 17, 223–224.
23. Hinderer, K. (1980): Grundbegriffe der Wahrscheinlichkeitstheorie. Springer-Verlag.Google Scholar
24. Hürlimann, W. (1995): Transforming, ordering and rating risks. Bulletin of the Swiss Association of Actuaries, 213–236.Google Scholar
25. Hürlimann, W. (1997a): Fonctions extrémales et gain financier. Elemente der Mathematik 52, 152–168.
26. Hürlimann, W. (1997b): Truncation transforms, stochastic orders and layer pricing. 26-th International Congress of Actuaries, June 1998, Birmingham.Google Scholar
27. Hürlimann, W. (1998a): On stop-loss order and the distortion pricing principle. ASTIN Bulletin 28, 119–134.
28. Hürlimann, W. (1998b): Higher degree stop-loss transforms and stochastic orders (II) applications.Google Scholar
29. Jean, W. H. (1980): The geometric mean and stochastic dominance. Journal of Finance 35, 151–158.
30. Kaas, R., van Heerwaarden, A. E., andGoovaerts, M. J. (1994): Ordering of Actuarial Risks. CAIRE Education Series 1, Brussels.Google Scholar
31. Kaas, R. andHesselager, O. (1995): Ordering claim size distributions and mixed Poisson probabilities. Insurance: Mathematics and Economics 17, 193–201.
32. Karlin, S. andNovikoff, A. (1963): Generalized convex inequalities. Pacific Journal of Mathematics 13, 1251–1279.
33. Keilson, J. andSumita, U. (1982): Unfirom stochastic and related inequalities. Canadian Journal of Statistics 10, 181–198.
34. Laffont, J. J. andTirole, J. (1993): A Theory of Incentives in Procurement and Regulation. Cambridge, MA: MIT Press.Google Scholar
35. Mosler, K. C. andScarsini, M. (1993): Stochastic orders and applications: a classified bibliography. Lecture notes in economics and mathematical systems, 401.Google Scholar
36. Müller, A. (1996): Ordering of risks: a comparative study via stop-loss transforms. Insurance: Mathematics and Economics 17, 215–222.
37. Muth, E. J. (1977): Reliability models with positive memory derived from the men residual life function. In Tsokas, C. P. and Shimi, I. N. (Eds.). The Theory and Applications of Reliability, vol. II, 401–434. Academic Press, New York.Google Scholar
38. O’Brien, G. L. (1984): Stochastic dominance and moment inequalities. Mathematics of Operations Research 9(3), 475–477.
39. Rolski, T. (1976): Order relations in the set of probability distribution functions and their applications to queuing theory. Dissertationes CXXXII. Warsaw.Google Scholar
40. Rothschild, M. andStiglitz, J. E. (1970): Increasing risk: I. A definition. Journal of Economic Theory 2, 225–243.
41. Rothschild, M. andStiglitz, J. E. (1971): Increasing risk: II. Its economic consequences. Journal of Economic Theory 3, 66–84.
42. Schröder, M. (1996): The Value at Risk Approach — Proposals on a Generalization. Aktuarielle Ansätze für Finanz-Risiken, AFIR 1996, vol. 1, 151–170. Verlag Versicherungswirtschaft, Karlsruhe.Google Scholar
43. Shaked, M. andShanthikumar, J. G. (1994): Stochastic orders and their applications. Academic Press, New York.
44. Smith, W. L. (1959): On the cumulants of renewal processes. Biometrika 46, 1–29.
45. Stoyan, D. (1977): Qualitative Eigenschaften und Abschätzungen stochastischer Modelle. Akademie-Verlag, Berlin. (English version (1983.) Comparison Methods for Queues and Other Stochastic Models. J. Wiley, New York.)Google Scholar
46. Szekli, R. (1995): Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics 97. Springer-Verlag.Google Scholar
47. Whitmore, G. A. (1970): Third degree stochastic dominance. American Economic Review 60, 457–459.Google Scholar
48. Whitt, W. (1980): The effect of variability in the GI/G/s queue. Journal of Applied Probability 17, 1062–1071.

## Authors and Affiliations

• Werner Hürlimann
• 1
1. 1.Winterthur