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On multiply monotone distributions, continuous or discrete, with applications

Author

Listed:
  • Claude Lefèvre

    (ULB - Département de Mathématique [Bruxelles] - ULB - Faculté des Sciences [Bruxelles] - ULB - Université libre de Bruxelles)

  • Stéphane Loisel

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer order t. A characterization is presented as a mixture of a minimum of t independent uniform distributions. Then, a comparison of t-monotone distributions is made using the s-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.

Suggested Citation

  • Claude Lefèvre & Stéphane Loisel, 2013. "On multiply monotone distributions, continuous or discrete, with applications," Post-Print hal-00750562, HAL.
    • Handle: RePEc:hal:journl:hal-00750562
    Note: View the original document on HAL open archive server: https://hal.science/hal-00750562v2
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    References listed on IDEAS

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    1. Denuit, Michel & Lefevre, Claude & Mesfioui, M'hamed, 1999. "On s-convex stochastic extrema for arithmetic risks," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 143-155, November.
    2. Lefèvre, Claude & Loisel, Stéphane, 2010. "Stationary-excess operator and convex stochastic orders," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 64-75, August.
    3. Edward Furman & Ričardas Zitikis, 2009. "Weighted Pricing Functionals With Applications to Insurance," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(4), pages 483-496.
    4. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, December.
    5. Denuit, Michel & Lefevre, Claude, 1997. "Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences," Insurance: Mathematics and Economics, Elsevier, vol. 20(3), pages 197-213, October.
    6. Goovaerts, Marc J. & Kaas, Rob & Dhaene, Jan & Tang, Qihe, 2003. "A Unified Approach to Generate Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 33(2), pages 173-191, November.
    7. Denuit, Michel & Vylder, Etienne De & Lefevre, Claude, 1999. "Extremal generators and extremal distributions for the continuous s-convex stochastic orderings," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 201-217, May.
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    Citations

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    Cited by:

    1. Anna Castañer & M. Mercè Claramunt, 2019. "Equilibrium Distributions and Discrete Schur-constant Models," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 449-459, June.
    2. Giguelay, J. & Huet, S., 2018. "Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population," Computational Statistics & Data Analysis, Elsevier, vol. 127(C), pages 96-115.
    3. Denuit, Michel & Liu, Liqun & Meyer, Jack, 2014. "A separation theorem for the weak s-convex orders," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 279-284.
    4. Claude Lefèvre & Stéphane Loisel & Pierre Montesinos, 2020. "Bounding Basis-Risk Using s-convex Orders on Beta-unimodal Distributions," Post-Print hal-02611227, HAL.
    5. Castañer, A. & Claramunt, M.M. & Lefèvre, C. & Loisel, S., 2015. "Discrete Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 343-362.
    6. Balabdaoui, Fadoua & Kulagina, Yulia, 2020. "Completely monotone distributions: Mixing, approximation and estimation of number of species," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    7. Castañer, A. & Claramunt, M.M. & Lefèvre, C. & Loisel, S., 2015. "Discrete Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 343-362.
    8. Castañer, Anna & Claramunt, M. Mercè & Lefèvre, Claude & Loisel, Stéphane, 2019. "Partially Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 172(C), pages 47-58.
    9. Anna Castañer & M Mercè Claramunt, 2017. "Equilibrium distributions and discrete Schur-constant models," Working Papers hal-01593552, HAL.
    10. Denuit, Michel, 2016. "Risk Apportionment and Multiply Monotone Targets," LIDAM Discussion Papers ISBA 2016044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Anna Casta~ner & M Merc`e Claramunt, 2017. "Equilibrium distributions and discrete Schur-constant models," Papers 1709.09955, arXiv.org.
    12. Peyhardi, Jean & Fernique, Pierre & Durand, Jean-Baptiste, 2021. "Splitting models for multivariate count data," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    13. Denuit, Michel M., 2018. "Risk apportionment and multiply monotone targets," Mathematical Social Sciences, Elsevier, vol. 92(C), pages 74-77.

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