Skip to main content
×
×
Home

Multivariate Convex Orderings, Dependence, and Stochastic Equality

  • Marco Scarsini (a1)
Abstract

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

Copyright
Corresponding author
Postal address: Dipartimento di Scienze, Università D'Annunzio, Viale Pindaro 42, I-65127 Pescara, Italy. e-mail address: scarsini@sci.unich.it
References
Hide All
Arnold, B.C. (1987). Majorization and the Lorenz Order: A Brief Introduction. (Lecture Notes in Statistics.) Springer, New York.
Baccelli, F., and Makowski, A.M. (1989). Multidimensional stochastic ordering and associated random variables. Operat. Res. 37, 478487.
Bhandari, S.K. (1988). Multivariate majorization and directional majorization; positive results. Sankhyā A 50, 199204.
Bhattacharjee, M.C. (1991). Some generalized variability orderings among life distributions with reliability applications. J. Appl. Prob. 28, 374383.
Bhattacharjee, M.C., and Sethuraman, J. (1990). Families of life distributions characterized by two moments. J. Appl. Prob. 27, 720725.
Chang, C.-S., Chao, X.L., Pinedo, M., and Shantikumar, J. G. (1991). Stochastic convexity for multi-dimensional processes and its applications. IEEE Trans. Automatic Control 36, 13471355.
Fishburn, P.C. (1980). Stochastic dominance and moments of distributions. Math. Operat. Res. 5, 94100.
Fishburn, P.C., and Lavalle, I.H. (1995). Stochastic dominance on unidimensional grids. Math. Operat. Res. 20, 513525.
Joe, H., and Verducci, J. (1993). Multivariate majorization by positive combinations. In Stochastic Inequalities. ed. Shaked, M. and Tong, Y. L. IMS Lecture Notes/Monograph Series, Hayward, CA. pp. 159181.
Johansen, S. (1972). A representation theorem for a convex cone of quasi convex functions. Math. Scand. 30, 297312.
Johansen, S. (1974). The extremal convex functions. Math. Scand. 34, 6168.
Koshevoy, G. (1995). Multivariate Lorenz majorization. Social Choice and Welfare 12, 93102.
Koshevoy, G. (1996). Lorenz zonotope and multivariate majorization. To appear in Social Choice and Welfare
Koshevoy, G., and Mosler, K. (1995). A geometrical approach to compare the variability of random vectors. Preprint. Universität der Bundeswehr Hamburg.
Koshevoy, G., and Mosler, K. (1996). The Lorenz zonoid of a multivariate distribution. J. Amer. Statist. Assoc. 91, 873882.
Li, H., and Zhu, H. (1994). Stochastic equivalence of ordered random variables with applications in reliability theory. Statist. Prob. Lett. 20, 383393.
Marshall, A.W., and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
Meester, L.E., and Shanthikumar, J.G. (1993). Regularity of stochastic processes: A theory based on directional convexity. Prob. Eng. Inf. Sci. 7, 343360.
Mosler, K., and Scarsini, M. (1991). Some theory of stochastic dominance. In Stochastic Orderings and Decision under Risk. ed. Mosler, K. and Scarsini, M. IMS Lecture Notes/Monograph Series, Hayward, CA. pp. 261284.
Mosler, K., and Scarsini, M. (1993). Stochastic Orders and Applications: A Classified Bibliography. (Lecture Notes in Economics and Mathematical Systems.) Springer, Berlin.
O'Brien, G.L. (1984). Stochastic dominance and moment inequalities. Math. Operat. Res. 9, 475477.
O'Brien, G.L., and Scarsini, M. (1991). Multivariate stochastic dominance and moments. Math. Operat. Res. 16, 382389.
Rüschendorf, L. (1980). Inequalities for the expectation of Δ-monotone functions. Z. Wahrscheinlichkeitsth. 54, 341349.
Sampson, A.R., and Whitaker, L.R. (1988). Positive dependence, upper sets, and multi-dimensional partitions. Math. Operat. Res. 13, 254264.
Scarsini, M. (1988). Multivariate stochastic dominance with fixed dependence structure. Operat. Res. Lett. 7, 237240.
Scarsini, M., and Shaked, M. (1990). Some conditions for stochastic equality. Nav. Res. Logist. 37, 617625.
Shaked, M., and Shanthikumar, J.G. (1994). Stochastic Orders and Their Applications. Academic Press, New York.
Tong, Y.L. (1990). The Multivariate Normal Distribution. Springer, New York.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification