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Two Variability Orders

Published online by Cambridge University Press:  27 July 2009

Moshe Shaked  and
J. George Shanthikumar
Moshe Shaked
Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721
J. George Shanthikumar
Affiliation:
School of Business Administration, University of California, Berkeley, California 94720
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Abstract

In this paper we study a new variability order that is denoted by ≤st:icx. This order has important advantages over previous variability orders that have been introduced and studied in the literature. In particular, X ≤st:icxY implies that Var[h(X)] ≤ Var[h (Y)] for all increasing convex functions h. The new order is also closed under formations of increasing directionally convex functions; thus it follows that it is closed, in particular, under convolutions. These properties make this order useful in applications. Some sufficient conditions for X ≤st:icxY are described. For this purpose, a new order, called the excess wealth order, is introduced and studied. This new order is based on the excess wealth transform which, in turn, is related to the Lorenz curve and to the TTT (total time on test) transform. The relationships to these transforms are also studied in this paper. The main closure properties of the order ≤st:icx are derived, and some typical applications in queueing theory are described.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 12 , Issue 1 , January 1998 , pp. 1 - 23
Copyright
Copyright © Cambridge University Press 1998

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References

1.Arnold, B.C. (1987). Majorization and the Lorenz order: A brief introduction. Berlin: Springer-Verlag.Google Scholar
2.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing: Probability models. New York: Holt, Rinehart and Winston.Google Scholar
3.Brown, M. & Shanthikumar, J.G. (1996). Comparing the variability of random variables and point processes. Technical Report, Department of Industrial Engineering and Operations Research, University of California, Berkeley.Google Scholar
4.Esary, J.D., Proschan, F., & Walkup, D.W. (1967). Association of random variables with applications. Annals of Mathematical Statistics 38: 1466–1474.CrossRefGoogle Scholar
5.Fagiuoli, E., Pellerey, F., & Shaked, M. (1997). A characterization of the dilation order and its applications. Technical Report, Department of Mathematics, University of Arizona.Google Scholar
6.Fernandez-Ponce, J.M., Kochar, S.C., & Munoz-Perez, J. (1997). Partial ordering of distributions based on right-spread functions. Journal of Applied Probability (to appear).Google Scholar
7.Jewitt, I. (1989). Choosing between risky prospects: The characterization of comparative statics results, and location independent risk. Management Science 35: 60–70.Google Scholar
8.Kochar, S.C. & Carrière, K.C. (1997). Connections among various variability orderings. Statistics and Probability Letters (to appear).Google Scholar
9.Landsberger, M. & Meilijson, I. (1994). The generating process and an extension of Jewitt's location independent risk concept. Management Science 40: 662–669.CrossRefGoogle Scholar
10.Lehmann, E.L. (1966). Some concepts of dependence. Annals of Mathematical Statistics 37: 1137–1153.CrossRefGoogle Scholar
11.Li, H., Scarsini, M., & Shaked, M. (1996). Linkages: A tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals. Journal of Multivariate Analysis 56: 20–41.CrossRefGoogle Scholar
12.Meester, L.E. & Shanthikumar, J.G. (1993). Regularity of stochastic processes: A theory based on directional convexity. Probability in the Engineering and Informational Sciences 7: 343–360.Google Scholar
13.Norros, I. (1986). A compensator representation of multivariate life length distributions, with applications. Scandinavian Journal of Statistics 13: 99–112.Google Scholar
14.Pham, T.G. & Turkkan, N. (1994). The Lorenz and the scaled total-time-on-test transform curves: A unified approach. IEEE Transactions on Reliability 43: 76–84.Google Scholar
15.Rubinstein, R.Y., Samorodnitsky, G., & Shaked, M. (1985). Antithetic variates, multivariate dependence and simulation of complex systems. Management Science 31: 66–77.Google Scholar
16.Rüschendorf, L. & de Valk, V. (1993). On regression representations of stochastic processes. Stochastic Processes and Their Applications 46: 183–198.CrossRefGoogle Scholar
17.Scarsini, M. (1994). Comparing risk and risk aversion. In Shaked, M. & Shanthikumar, J.C. (eds.), Stochastic orders and their applications. Boston: Academic Press, pp. 351–378.Google Scholar
18.Shaked, M. & Shanthikumar, J.G. (1986). The total hazard construction, antithetic variates and simulation of stochastic systems. Communications in Statistics—Stochastic Models 2: 237–249.CrossRefGoogle Scholar
19.Shaked, M. & Shanthikumar, J.G. (1987). The multivariate hazard construction. Stochastic Processes and Their Applications 24: 241–258.CrossRefGoogle Scholar
20.Shaked, M. & Shanthikumar, J.G. (1990). Parametric stochastic convexity and concavity of stochastic processes. Annals of the Institute of Statistical Mathematics 42: 509—531.Google Scholar
21.Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. Boston: Academic Press.Google Scholar
22.Shanthikumar, J.G. & Yao, D.D. (1989). Second-order stochastic properties in queueing systems. Proceedings of the IEEE 77: 162–170.CrossRefGoogle Scholar
23.Shanthikumar, J.G. & Yao, D.D. (1991). Strong stochastic convexity: Closure properties and applications. Journal of Applied Probability 28: 131–145.CrossRefGoogle Scholar