Fréchet Classes and Nonmonotone Dependence

  • Marco Scarsini
  • Moshe Shaked


The analysis of the Fréchet class of bivariate distribution functions with fixed marginals is fundamental in the study of positive dependence. Several notions of positive dependence have been examined in the literature (see (1982), Kimeldorf and Sampson (1987, Kimeldorf and Sampson 1989), and (1996) for surveys of the field). All of these notions tend to capture the idea that, for a certain random pair, large values of one random variable go together with large values of the other random variable. Another way of expressing this concept is to say that the distribution of the random pair tends to concentrate its mass around the graph of an increasing function. The stronger the dependence, the more concentrated the probability mass will be around the graph of the function. If the marginal distributions are continuous, in the extreme case of a perfect positive dependence all the mass will lie exactly on the graph of an increasing function. Since the marginals are fixed, there exists only one increasing function on whose graph the probability mass can concentrate.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahmed, A.-H. N., León, R., and Proschan, F. (1981), Generalized association, with applications in multivariate statistics, Annals of Statistics 9, 168–176.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Dabrowska, D. (1985), Descriptive parameters of location, dispersion and stochastic dependence, Statistics 16, 63–88.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Dellacherie, C. (1971), Quelques commentaires sur les prolongements de capacités, Séminaire de Probabilités V, Lecture Notes in Mathematics 191, Springer-Verlag, Berlin, 77–81.Google Scholar
  4. Fréchet, M. (1951), Sur les tableaux de corrélation dont les marges sont données, Annales de l’Université de Lyon 9, sec A., 53–77.Google Scholar
  5. Kimeldorf, G. and Sampson, A. R. (1987), Positive dependence orderings, Annals of the Institute of Statistical Mathematics 39, 113–128.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Kimeldorf, G. and Sampson, A. R. (1989), A framework for positive dependence, Annals of the Institute of Statistical Mathematics 41, 31–45.MathSciNetzbMATHGoogle Scholar
  7. Scarsini, M. (1989), Copulae of probability measures on product spaces, Journal of Multivariate Analysis 31, 201–219.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Scarsini, M. and Shaked, M. (1996), Positive dependence orders: A survey, Athens Conference on Applied Probability and Time Series I: Applied Probability, (Eds: C. C. Heyde, Y. V. Prohorov, R. Pyke and S. T. Rachev), Lecture Notes in Statistics 114, Springer-Verlag, Berlin, 70–91.CrossRefGoogle Scholar
  9. Scarsini, M. and Venetoulias, A. (1993), Bivariate distributions with nonmonotone dependence structure, Journal of the American Statistical Association 88, 338–344.MathSciNetzbMATHGoogle Scholar
  10. Schweizer, B. and Sklar, A. (1983), Probabilistic Metric Spaces, Elsevier, New York.zbMATHGoogle Scholar
  11. Shaked, M. (1982), A general theory of some positive dependence notions, Journal of Multivariate Analysis 12, 199–218.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Marco Scarsini
    • 1
  • Moshe Shaked
    • 2
  1. 1.Dipartimento di ScienzeUniversità D’AnnunzioPescaraItaly
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA

Personalised recommendations