## Journal of Applied Probability

### On multiply monotone distributions, continuous or discrete, with applications

#### 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.

#### Article information

Source
J. Appl. Probab., Volume 50, Number 3 (2013), 827-847.

Dates
First available in Project Euclid: 5 September 2013

https://projecteuclid.org/euclid.jap/1378401239

Digital Object Identifier
doi:10.1239/jap/1378401239

Mathematical Reviews number (MathSciNet)
MR3102517

Zentralblatt MATH identifier
1293.62028

#### Citation

Lefèvre, Claude; Loisel, Stéphane. On multiply monotone distributions, continuous or discrete, with applications. J. Appl. Probab. 50 (2013), no. 3, 827--847. doi:10.1239/jap/1378401239. https://projecteuclid.org/euclid.jap/1378401239

#### References

• Abramowitz, M. and Stegun, I. A. (eds) (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
• Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
• Balabdaoui, F. and Wellner, J. A. (2007). Estimation of a $k$-monotone density: limit distribution theory and the spline connection. Ann. Statist. 35, 2536–2564.
• Bertin, E. M. J., Cuculescu, I. and Theodorescu, R. (1997). Unimodality of Probability Measures. Kluwer, Dordrecht.
• Constantinescu, C., Hashorva, E. and Ji, L. (2011). Archimedian copulas in finite and infinite dimensions–-with applications to ruin problems. Insurance Math. Econom. 49, 487–495.
• Cox, D. R. (1962). Renewal Theory. John Wiley, New York.
• De Jong, P. and Madan, D. B. (2011). Capital adequacy of financial enterprises. Working paper. Available at http://ssrn.com/abstract=1761107.
• Denuit, M. and Lefèvre, C. (1997). Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences. Insurance Math. Econom. 20, 197–213.
• Denuit, M., De Vylder, E. and Lefèvre, C. (1999a). Extremal generators and extremal distributions for the continuous $s$-convex stochastic orderings. Insurance Math. Econom. 24, 201–217.
• Denuit, M., Lefèvre, C. and Mesfioui, M. (1999b). On $s$-convex stochastic extrema for arithmetic risks. Insurance Math. Econom. 25, 143–155.
• Denuit, M., Lefèvre, C. and Shaked, M. (1998). The $s$-convex orders among real random variables, with applications. Math. Inequal. Appl. 1, 585–613.
• Denuit, M., Lefèvre, C. and Shaked, M. (2000). Stochastic convexity of the Poisson mixture model. Methodology Comput. Appl. Prob. 2, 231–254.
• Denuit, M., Lefèvre, C. and Utev, S. (1999c). Generalized stochastic convexity and stochastic orderings of mixtures. Prob. Eng. Inf. Sci. 13, 275–291.
• Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
• Furman, E. and Zitikis, R. (2009). Weighted pricing functionals with applications to insurance: an overview. N. Amer. Actuarial J. 13, 483–496.
• Gerber, H. U. (1972). Ein satz von Khintchin und die varianz von unimodalen. Bull. Swiss Assoc. Actuaries, 225–231.
• Gneiting, T. (1999). Radial positive definite functions generated by Euclid's hat. J. Multivariate Anal. 69, 88–119.
• Goovaerts, M. J., Kaas, R., Dhaene, J. and Tang, Q. (2003). A unified approach to generate risk measures. ASTIN Bull. 33, 173–192.
• Goovaerts, M. J., Kaas, R., Van Heerwaarden, A. E. and Bauwelinckx, T. (1990). Effective Actuarial Methods. North-Holland, Amsterdam.
• Kaas, R. and Goovaerts, M. J. (1987). Unimodal distributions in insurance. Bull. Assoc. R. Actuaires Belges 81, 61–66.
• Kaas, R., van Heerwaarden, A. E. and Goovaerts, M. J. (1994). Ordering of Actuarial Risks. CAIRE, Brussels.
• Kaas, R., Goovaerts, M. J., Dhaene, J. and Denuit, M. (2008). Modern Actuarial Risk Theory: Using R. Springer, Heidelberg.
• Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. John Wiley, New York.
• Lefèvre, C. and Loisel, S. (2010). Stationary-excess operator and convex stochastic orders. Insurance Math. Econom. 47, 64–75.
• Lefèvre, C. and Utev, S. (1996). Comparing sums of exchangeable Bernoulli random variables. J. Appl. Prob. 33, 285–310.
• Lefèvre, C. and Utev, S. (2013). Convolution property and exponential bounds for symmetric monotone densities. ESAIM Prob. Statist. 17, 605–613.
• Lévy, P. (1962). Extensions d'un théorème de D. Dugué et M. Girault. Z. Wahrscheinlichkeitsth. 1, 159–173.
• Pakes, A. G. (1996). Length biasing and laws equivalent to the log-normal. J. Math. Anal. Appl. 197, 825–854.
• Pakes, A. G. (1997). Characterization by invariance under length-biasing and random scaling. J. Statist. Planning Infer. 63, 285–310.
• Pakes, A. G. (2003). Biological applications of branching processes. In Stochastic Processes: Modelling and Simulation (Handbook Statist. 21), eds D. N. Shanbhag and C. R. Rao, North-Holland, Amsterdam, pp. 693–773.
• Pakes, A. G. and Navarro, J. (2007). Distributional characterizations through scaling relations. Austral. N. Ze. J. Statist. 49, 115–135.
• Patil, G. P. and Rao, C. R. (1978). Weighted distributions and size-biased sampling with applications to wildlife populations and human families. Biometrics 34, 179–189.
• Pečarić, J. E., Proschan, F. and Tong, Y. L. (1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, Boston, MA.
• Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
• Steutel, F. W. and van Harn, K. (1979). Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893–899.
• Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23, 189–207. \endharvreferences