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Blätter der DGVFM

, Volume 24, Issue 3, pp 449–463 | Cite as

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.

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References

  1. Ali, M. M. (1974): Stochastic ordering and kurtosis measure. Journal of the American Statistical Association 69, 543–545.MATHCrossRefMathSciNetGoogle Scholar
  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.MATHGoogle Scholar
  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.MATHCrossRefMathSciNetGoogle Scholar
  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.CrossRefMathSciNetGoogle Scholar
  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.CrossRefGoogle Scholar
  12. Fishburn, P. C. (1980): Stochastic dominance and moments of distributions. Mathematics of Operations Research 5, 94–100.MATHCrossRefMathSciNetGoogle Scholar
  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.CrossRefMathSciNetGoogle Scholar
  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.MATHGoogle Scholar
  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.MATHCrossRefGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  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.MATHCrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  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.CrossRefMathSciNetGoogle Scholar
  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.MATHCrossRefMathSciNetGoogle Scholar
  32. Karlin, S. andNovikoff, A. (1963): Generalized convex inequalities. Pacific Journal of Mathematics 13, 1251–1279.MATHMathSciNetGoogle Scholar
  33. Keilson, J. andSumita, U. (1982): Unfirom stochastic and related inequalities. Canadian Journal of Statistics 10, 181–198.MATHCrossRefMathSciNetGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  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.MATHCrossRefMathSciNetGoogle Scholar
  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.CrossRefMathSciNetGoogle Scholar
  41. Rothschild, M. andStiglitz, J. E. (1971): Increasing risk: II. Its economic consequences. Journal of Economic Theory 3, 66–84.CrossRefMathSciNetGoogle Scholar
  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.MATHGoogle Scholar
  44. Smith, W. L. (1959): On the cumulants of renewal processes. Biometrika 46, 1–29.MATHMathSciNetGoogle Scholar
  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.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© DAV/DGVFM 2000

Authors and Affiliations

  • Werner Hürlimann
    • 1
  1. 1.Winterthur

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