Advertisement

Multivariate tests of independence

  • Paul Deheuvels
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 861)

Abstract

If Xn=(Xn(1),...,Xn(p)), n=1,2,... is an i.i.d. sequence of p-variate random vectors, we consider here the problem of distribution free hypothesis testing on the probability law of Xn, that is, to test hypothesis invariant under continuous non decreasing transforms of the coordinates. Our main interest will be for tests of independence, typical among the preceding.

A fundamental tool for the study of these problems appears to be the dependence functions of which we summarize the main definitions and basic results in §1. Doing this, we introduce the new dependogram statistic which has proved itself to be an efficient way to visualize the information contained in a sample on its intern dependence structure.

We assume afterwords that the hypothesis of independence is true, and state in §2 some important results about the weak functional convergence of the normalized empirical dependence function toward a limiting Gaussian process.

As an application of the preceding, we derive in §3 the main original result of this paper, obtaining a general family of Open image in new window statistical tests for independence extending results proved for the bivariate case (p=2) in [2] and [4].

We give in §4 an explicit example of such a statistic for the trivariate (p=3) case. We obtain a Open image in new window (4) as a limiting distribution, and give a partition of this test in four asymptotically independent and Open image in new window (1) components.

In §5, we give an extension of a result proved in [5] about Kolmogorov-Smirnov tests of independence in the bivariate case (p=2).

Keywords

Dependence Function Multivariate Test Empirical Dependence Bivariate Case Centered Gaussian Process 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    DEHEUVELS P., 1978, Caractérisation complète des lois extrêmes multivariées et de la convergence des types extrêmes, Publ. ISUP f.3–4, p. 1–37.Google Scholar
  2. [2]
    , 1979, La fonction de dépendance empirique et ses propriétés, un test non paramétrique d’indépendance, Acad. Royale de Belgique, Bull. Classe des Sciences, 5e ser. LXV, 6, p. 274–292.MathSciNetzbMATHGoogle Scholar
  3. [3]
    1980, The decomposition of infinite order and extreme multivariate distributions, in Asymptotic theory of statistical tests and estimation, Chakravarti edit., Acad. Press.Google Scholar
  4. [4]
    1980, Nonparametric tests of independence, in Actes du Colloque de Rouen, Raoult edit., Springer Verlag lecture notes.Google Scholar
  5. [5]
    1981, A Kolmogorov-Smirnov test for independence, accepted for publication in Revue Roum. Math. Pures Appl.Google Scholar
  6. [6]
    1981, Some applications of the dependence functions to statistical inference, Coll. Math. Janos Bolyai, 1980, Vincze edit., North Holland.Google Scholar
  7. [7]
    1981, An asymptotic decomposition for multivariate tests of independence, Accepted for publication in the Journal of Multivariate Analysis.Google Scholar
  8. [8]
    EVERITT B.S., 1977, The analysis of contingency tables, Chapman & Hall.Google Scholar
  9. [9]
    FRECHET M., 1951, Sur les tableaux de corrélation dont les marges sont données, Ann. Univ. Lyon, 3e Ser., Ser. A, p. 53–77.Google Scholar
  10. [10]
    GALAMBOS J., 1978, The asymptotic theory of extreme order statistics, Wiley.Google Scholar
  11. [11]
    PURI M.L., SEN P.K., 1971, Nonparametric methods in multivariate analysis, Wiley.Google Scholar
  12. [12]
    RUSCHENDORF L., 1976, Asymptotic distributions of multivariate rank statistics, Ann. Statist. 4, p. 912–923.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    SKLAR A., 1973, Random variables, joint distribution functions and copulas, Kybernetika 9.Google Scholar
  14. [14]
    SIBUYA M., 1959, Bivariate extreme statistics, Ann. Inst. Stat. Math., 11, p. 199–210.MathSciNetCrossRefGoogle Scholar
  15. [15]
    TIAGO DE OLIVEIRA J., 1975, Bivariate and multivariate extreme distributions, in Statistical distributions in scientific work, Patil, Kotz & Ord edit., Reidel.Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Paul Deheuvels
    • 1
    • 2
  1. 1.Bourg-la-ReineFrance
  2. 2.I.S.U.P., t. 45–55, 3e ét.Université Paris VIParis Cedex 05

Personalised recommendations