Some Concepts of Dependence

  • E. L. Lehmann
Open Access
Part of the Selected Works in Probability and Statistics book series (SWPS)

Summary and introduction

Problems involving dependent pairs of variables (X, Y) have been studied most intensively in the case of bivariate normal distributions and of 2 × 2 tables. This is due primarily to the importance of these cases but perhaps partly also to the fact that they exhibit only a particularly simple form of dependence. (See Examples 9(i) and 10 in Section 7.) Studies involving the general case center mainly around two problems: (i) tests of independence; (ii) definition and estimation of measures of association. In most treatments of these problems, there occurs implicitly a concept which is of importance also in other contexts (for example, the evaluation of the performance of certain multiple decision procedures), the concept of positive (or negative) dependence or association. Tests of independence, for example those based on rank correlation, Kendall’s Z-statistic, or normal scores, are usually not omnibus tests (for a discussion of such tests see [4], [15] and [17], but designed to detect rather specific types of alternatives, namely those for which large values of Y tend to be associated with large values of X and small values of Y with small values of X (positive dependence) or the opposite case of negative dependence in which large values of one variable tend to be associated with small values of the other. Similarly, measures of association are typically designed to measure the degree of this kind of association. The purpose of the present paper is to give three successively stronger definitions of positive dependence, to investigate their consequences, explore the strength of each definition through a number of examples, and to give some statistical applications.


Conditional Distribution Nondecreasing Function Multinomial Distribution Quadrant Dependence Bivariate Normal Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bhuchongkul, S. (1964). A class of nonparametric tests for independence in bivariate populations. Ann. Math. Statist. 35 138–149.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bickel, Petek J. (1965). Some contributions to the theory of order statistics. Submitted.Google Scholar
  3. 3.
    Blomqvist, Nils (1950). On a measure of dependence between two random variables. Ann. Math. Statist. 21 593–600.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Blum, J. R., Kiefer, J. and Rosenblatt, M. (1961). “Distribution free tests of independence based on the sample distribution,” Ann. Math. Statist. 32 485–498.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cochran, W. G. (1941). The distribution of the largest of a set of estimated variances as a fraction of their total. Ann. Eugenics 11 47–52.MathSciNetCrossRefGoogle Scholar
  6. 6.
    David, H. A. (1956). On the application to statistics of an elementary theorem in probability. Biometrika 43 85–91.MathSciNetMATHGoogle Scholar
  7. 7.
    Doornbos, R. (1956). Significance of the smallest of a set of estimated normal variances. Statistica Neerlandica, 10 17–26.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Doornbos, R. and Prins, H. J. (1958). On slippage tests. Indag. Math. 20 38–55 and 438–447.MathSciNetGoogle Scholar
  9. 9.
    Efron, Bradley (1965). Increasing properties of Pó1ya frequency functions. Ann. Math. Statist. 36 272–279.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Halperin, M., Greenhouse, S. W., Cornfield, J. and Zalokar, Julia (1955). Tables of percentage points for the studentifed maximum absolute deviate in normal samples. J. Amer. Statist. Assoc. 50 185–195.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934). Inequalities. Cambridge Univ. Press.Google Scholar
  12. 12.
    Hartley, H. O. (1955). Some recent developments in analysis of variance. Comm. Pure Appl. Math. 8 47–72.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Höffding, Wassily (1940). Masstabinvariante Korrelations-theorie. Schriften Math. Inst. Univ. Berlin 5 181–233.Google Scholar
  14. 14.
    Höeffding, Wassily (1948). A class of statistics with asymptotically normal distributions. Ann. Math. Statist. 19 293–375.MATHCrossRefGoogle Scholar
  15. 15.
    Hö effding, Wassily (1948). A nonparametric test of independence. Ann. Math. Statist. 19 546–557.MATHCrossRefGoogle Scholar
  16. 16.
    Kruskal, William H. (1958). Ordinal measures of association. J. Amer. Statist. Assoc. 53 814–861.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Lehmann, E. L. (1951). Consistency and unbiasedness of certain nonparametric tests. Ann. Math. Statist. 22 165–179.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lehmann, E. L. (1959). Testing statistical hypotheses. Wiley, New York.MATHGoogle Scholar
  19. 19.
    Paulson, Edward (1952). On the comparison of several experimental categories with a control. Ann. Math. Statist. 23 239–246.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Paulson, Edward (1952). An optimum solution to the k-sample slippage problem for the normal distribution. Ann. Math. Statist. 23 610–616.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Tukey, J. W. (1958). A problem of Berkson, and minimum variance orderly estimators. Ann. Math. Statist. 29 588–592.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • E. L. Lehmann
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

Personalised recommendations