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On multiply monotone distributions, continuous or discrete, with applications


  • Claude Lefèvre

    () (ULB - Département de Mathématique [Bruxelles] - ULB - Université Libre de Bruxelles [Bruxelles])

  • Stéphane Loisel

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


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
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    References listed on IDEAS

    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. 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.
    4. 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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    Cited by:

    1. 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.
    2. Anna Casta~ner & M Merc`e Claramunt, 2017. "Equilibrium distributions and discrete Schur-constant models," Papers 1709.09955,
    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. Anna Castañer & M Mercè Claramunt, 2017. "Equilibrium distributions and discrete Schur-constant models," Working Papers hal-01593552, HAL.

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