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On a stochastic order induced by an extension of Panjer’s family of discrete distributions

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  • Aleksandr Beknazaryan

    (University of Twente)

  • Peter Adamic

    (Laurentian University)

Abstract

We factorize probability mass functions of discrete distributions belonging to Panjer’s family and to its certain extensions to define a stochastic order on the space of distributions supported on $${\mathbb {N}}_0$$ N 0 . Main properties of this order are presented. Comparison of some well-known distributions with respect to this order allows to generate new families of distributions that satisfy various recurrrence relations. The recursion formula for the probabilities of corresponding compound distributions for one such family is derived. Applications to various domains of reliability theory are provided.

Suggested Citation

  • Aleksandr Beknazaryan & Peter Adamic, 2022. "On a stochastic order induced by an extension of Panjer’s family of discrete distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(1), pages 67-91, January.
    • Handle: RePEc:spr:metrik:v:85:y:2022:i:1:d:10.1007_s00184-021-00822-5
    DOI: 10.1007/s00184-021-00822-5
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    References listed on IDEAS

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    2. Kitano, Masashi & Shimizu, Kunio & Ong, S.H., 2005. "The generalized Charlier series distribution as a distribution with two-step recursion," Statistics & Probability Letters, Elsevier, vol. 75(4), pages 280-290, December.
    3. Hesselager, Ole, 1994. "A Recursive Procedure for Calculation of some Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 24(1), pages 19-32, May.
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    5. Willmot, Gordon E., 1989. "Limiting tail behaviour of some discrete compound distributions," Insurance: Mathematics and Economics, Elsevier, vol. 8(3), pages 175-185, November.
    6. Sundt, Bjørn & Jewell, William S., 1981. "Further Results on Recursive Evaluation of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 27-39, June.
    7. Hess, Klaus Th. & Liewald, Anett & Schmidt, Klaus D., 2002. "An Extension of Panjer's Recursion," ASTIN Bulletin, Cambridge University Press, vol. 32(2), pages 283-297, November.
    8. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    9. Panjer, H. H. & Willmot, G. E., 1982. "Recursions for Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 13(1), pages 1-12, June.
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