Abstract
LetX andY be two random variables with finite expectationsE X andE Y, respectively. ThenX is said to be smaller thanY in the dilation order ifE[ϕ(X-E X)]≤E[ϕ(Y-E Y)] for any convex functionϕ for which the expectations exist. In this paper we obtain a new characterization of the dilation order. This characterization enables us to give new interpretations to the dilation order, and using them we identify conditions which imply the dilation order. A sample of applications of the new characterization is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, B. C. (1987).Majorization and the Lorenz Order: A Brief Introduction, Springer-Verlag, New York, NY.
Belzunce, F., Candel, J. andRuiz, J. M. (1996), Dispersive ordering and characterizations of ageing classes.Statistics and Probability Letters 28, 321–327.
Belzunce, F., Pellerey, F., Ruiz, J. M. andShaked, M. (1997), The dilation order, the dispersion order, and orderings of residual lives,Statistics and Probability Letters 33, 263–275.
Fernandez-Ponce, J. M., Kochar, S. C. andMuñoz-Perez, J. (1998), Partial ordering of distributions based on right spread functions,Journal of Applied Probability 35, 221–228.
Hickey, R. J. (1986), Concepts of dispersion in distributions: A comparative note,Journal of Applied Probability 23, 914–921.
Jewitt, I. (1989), Choosing between risky prospects: The characterization of comparative statics results, and location independent risk.Management Science 35, 60–70.
Kochar, S. C. (1989), On extensions of DMRL and related partial orderings of life distributions,Communications in Statistics—Stochastic Models,5, 235–245.
Kochar, S. C. andCarrière, K. C. (1997), Connections among various variability orderings,Statistics and Probability Letters,35, 327–333.
Landsberger, M. andMellijson, I. (1994), The generating process and an extension of Jewitt's location independent risk concept,Management Science 40, 662–669.
Levy, H. andKroll, Y. (1978), Ordering uncertain options with borrowing and lending,Journal of Finance 33, 553–574.
McDonald, J. B. andXu, Y. J. (1995a). A generalization of the beta distribution with applications,Journal of Econometrics 66, 133–152.
McDonald, J. B. andXu, Y. J. (1995b), Errata: A generalization of the beta distribution with applications.Journal of Econometrics 69, 427–428.
Moyes, P. (1987), A new concept of Lorenz domination,Economics Letters 23, 203–207.
Muñoz-Perez, J. andSanches-Gomez, A. (1990), Dispersive ordering by dilation,Journal of Applied Probability 27, 440–444.
Scarsini, M. (1994), Comparing risk and risk aversion, inStochastic Orders and their Applications (Eds: Shaked, M. and Shanthikumar, J. G.), Academic Press, New York, 351–378.
Shaked, M. andShanthikumar, J. G. (1994),Stochastic Orders and Their Applications, Academic Press, New York.
Shaked, M. andShanthikumar, J. G. (1998), Two variability orders,Probability in the Engineering and Informational Sciences 12, 1–23.
Shorrocks, A. F. (1987), Ranking income distributions,Economica 50, 3–17.
Taillie, C. (1981), Lorenz ordering within the generalized gamma family of distributions,Statistical Distributions in Scientific Work, Vol.6, (Eds: Taillie, C., Patil, G. P. and Baldessari, B. A.), D. Reidel, Boston, 181–192.
Wilfling, B. (1996), Lorenz ordering of generalized beta-II income distributions,Journal of Econometrics 71, 381–388.
Author information
Authors and Affiliations
Additional information
Partially supported by MURST 40% Program on Non-Linear Systems and Applications.
Partially supported by “Gruppo Nazionale per l'Analisi Funzionale e sue Applicazioni”—CNR.
Rights and permissions
About this article
Cite this article
Fagiuoli, E., Pellerey, F. & Shaked, M. A characterization of the dilation order and its applications. Statistical Papers 40, 393–406 (1999). https://doi.org/10.1007/BF02934633
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02934633