Inequalities for Ek(X, Y) when the marginals are fixed

  • Stamatis Cambanis
  • Gordon Simons
  • William Stout
Article

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.

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References

  1. 1.
    Adams, C. R., Clarkson, J. A.: Properties of functions f(x,y) of bounded variation. Trans. Amer. Math. Soc. 36, 711–730 (1934)Google Scholar
  2. 2.
    Bártfai, P.: über die Entfernung der Irrfahrtswege. Studia Sci. Math. Hungar. 5, 41–49 (1970)Google Scholar
  3. 3.
    Dall'Aglio, G.: Sugli estremi dei momenti delle funzioni di ripartizione doppia. Ann. Sci. école Norm. Sup. 10, 35–74 (1956)Google Scholar
  4. 4.
    Dall'Aglio, G.: Fréchet classes and compatibility of distribution functions. Symposia Mathematica 9, 131–150. New York: Academic Press 1972Google Scholar
  5. 5.
    Fréchet, M.: Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon, Sect. A, Sér. 3, 14, 53–77 (1951)Google Scholar
  6. 6.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge: Cambridge Univ. Press 1952Google Scholar
  7. 7.
    Hobson, E.W.: The Theory of Functions of a Real Variable and the Theory of Fourier's Series. Washington: Harren Press 1950Google Scholar
  8. 8.
    Hoeffding, W.: Ma\stabvariante Korrelationstheorie. Schr. Math. Inst. Univ. Berlin 5, 181–233 (1940)Google Scholar
  9. 9.
    Lehmann, E. L.: Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153 (1966)Google Scholar
  10. 10.
    Neumann, J. v.: Functional Operators. Volume I: Measures and Integrals. Princeton: Princeton Univ. Press 1950Google Scholar
  11. 11.
    Pledger, G., Proschan, F.: Stochastic comparisons of random processes, with applications in reliability. J. Appl. Probability 10, 572–585 (1973)Google Scholar
  12. 12.
    Skorokhod, A. V.: Limit theorems for stochastic processes. Theor. Probability Appl. 1, 261–290 (1965)Google Scholar
  13. 13.
    Tchen, A.H.T.: Exact inequalities for ∫Φ(x, y) dH(x, y) when H has given marginals. Manuscript (1975)Google Scholar
  14. 14.
    Vallender, S.S.: Calculation of the Wasserstein distance between probability distributions on the line. Theor. Probability Appl. 18, 784–786 (1973)Google Scholar
  15. 15.
    Veinott, A.F., Jr.: Oprimal policy in a dynamic, single product, nonstationary inventory model with several demand classes. Operations Res. 13, 761–778 (1965)Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Stamatis Cambanis
    • 1
  • Gordon Simons
    • 1
  • William Stout
    • 2
  1. 1.Department of StatisticsUniversity of North CarolinaChapel HillUSA
  2. 2.Department of MathematicsUniversity of IllinoisChampaign-UrbanaUSA

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