Abstract
When k(x, y) is a quasi-monotone function and the random variables X and Y have fixed distributions, it is shown under some further mild conditions that ℰ k(X, Y) is a monotone functional of the joint distribution function of X and Y. Its infimum and supremum are both attained and correspond to explicitly described joint distribution functions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, C. R., Clarkson, J. A.: Properties of functions f(x,y) of bounded variation. Trans. Amer. Math. Soc. 36, 711–730 (1934)
Bártfai, P.: über die Entfernung der Irrfahrtswege. Studia Sci. Math. Hungar. 5, 41–49 (1970)
Dall'Aglio, G.: Sugli estremi dei momenti delle funzioni di ripartizione doppia. Ann. Sci. école Norm. Sup. 10, 35–74 (1956)
Dall'Aglio, G.: Fréchet classes and compatibility of distribution functions. Symposia Mathematica 9, 131–150. New York: Academic Press 1972
Fréchet, M.: Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon, Sect. A, Sér. 3, 14, 53–77 (1951)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge: Cambridge Univ. Press 1952
Hobson, E.W.: The Theory of Functions of a Real Variable and the Theory of Fourier's Series. Washington: Harren Press 1950
Hoeffding, W.: Ma\stabvariante Korrelationstheorie. Schr. Math. Inst. Univ. Berlin 5, 181–233 (1940)
Lehmann, E. L.: Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153 (1966)
Neumann, J. v.: Functional Operators. Volume I: Measures and Integrals. Princeton: Princeton Univ. Press 1950
Pledger, G., Proschan, F.: Stochastic comparisons of random processes, with applications in reliability. J. Appl. Probability 10, 572–585 (1973)
Skorokhod, A. V.: Limit theorems for stochastic processes. Theor. Probability Appl. 1, 261–290 (1965)
Tchen, A.H.T.: Exact inequalities for ∫Φ(x, y) dH(x, y) when H has given marginals. Manuscript (1975)
Vallender, S.S.: Calculation of the Wasserstein distance between probability distributions on the line. Theor. Probability Appl. 18, 784–786 (1973)
Veinott, A.F., Jr.: Oprimal policy in a dynamic, single product, nonstationary inventory model with several demand classes. Operations Res. 13, 761–778 (1965)
Author information
Authors and Affiliations
Additional information
Research supported by the Air Force Office of Scientific Research under Grant AFOSR-75-2796
Research supported by the National Science Foundation
Rights and permissions
About this article
Cite this article
Cambanis, S., Simons, G. & Stout, W. Inequalities for E k(X, Y) when the marginals are fixed. Z. Wahrscheinlichkeitstheorie verw Gebiete 36, 285–294 (1976). https://doi.org/10.1007/BF00532695
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00532695