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On Multiply Monotone Distributions, Continuous or Discrete, with Applications

Part of: Applications Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Claude Lefèvre  and
Stéphane Loisel
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Stéphane Loisel*
Affiliation:
Université de Lyon
*
∗ Postal address: Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP 210, B-1050 Bruxelles, Belgium. Email address: clefevre@ulb.ac.be
∗∗ Postal address: Université de Lyon, Université Claude Bernard Lyon 1, I.S.F.A., 50 Avenue Tony Garnier, F-69007 Lyon, France. Email address: stephane.loisel@univ-lyon1.fr
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Abstract

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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.

Keywords

t-monotone functions-convex stochastic orderstationary-excess operatorMarkov and Lyapunov inequalitiesinsurance risk theory

MSC classification

Primary: 62E10: Characterization and structure theory 60E15: Inequalities; stochastic orderings
Secondary: 62P05: Applications to actuarial sciences and financial mathematics 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 827 - 847
Copyright
© Applied Probability Trust 

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