Non parametric tests of independence

  • Paul Deheuvels
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 821)


If (Xn)(1),...,Xn(p)) is for n=1,2,..., an i.i.d. sequence, with Fn(x1,...,xp) as its empirical c.d.f. with margins F n (j) , 1⩽j⩽p, the empirical dependence function Dn is the c.d.f. of a probability distribution with uniform margins on [0,1]p, and such that Fn(x1,...,xp)=Dn(F n (1) (x1),...,F n (p) (xp)). We show in this paper that Dn(u1,...,up) is asymptotically normal for p⩾3 and show the weak convergence of n1/2(Dn(u1,...,up)−E(Dn(u1,...,up))) toward a limiting gaussian process of which we derive the covariance function in the independence case. These results extend the bivariate case studied in [3] and [5].

Some applications are given to tests of independence, including in particular Kendall’s τ and Spearman’s ρ. We give a tabulation of our test Tn(4), developed in [3], for n=11−30, extending the tabulation for n=3−10 obtained in [4].

Key words

Non parametric methods tests of independence distribution free procedures rank statistics AMS(MOS) Classification Primary 62G10 Secundary 62G05 62E15 62E20 60G57 62Q05 


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  1. (1).
    BLUM, J.R., KIEFER, J., ROSENBLATT, M., 1961, Distribution free tests of independence based on the sample distribution function, Ann. Math. Statist., 32, pp. 485–497.MathSciNetCrossRefzbMATHGoogle Scholar
  2. (2).
    BHUCHONGKUL, S., 1964, A class of non parametric tests for independence in bivariate populations, Ann.Math. Statist., 35, p. 138–149.MathSciNetCrossRefzbMATHGoogle Scholar
  3. (3).
    DEHEUVELS P., 1979, La fonction de dépendance empirique et ses propriétés, un test non-paramétrique d’indépendance, Acad. Royale de Belgique, Bulletin de la classe des sciences, 5e Sér., t. LXV, f.6, p. 274–292.MathSciNetzbMATHGoogle Scholar
  4. (4).
    -, 1980, A non parametric test for independence, submitted to the Z. Wahrscheinlichkeit.Google Scholar
  5. (5).
    -, 1980, A Kolmogorov-Smirnov test for independence, to be published in Revue Roumaine de Math. Pures et appliquées.Google Scholar
  6. (6).
    -, 1980, Some applications of the dependence functions to statistical inference: non parametric estimates of extreme value distributions, and a Kiefer type universal bound for the uniform test of independence, to be published in Coll. Math. Janos Bolyai soc.Google Scholar
  7. (7).
    DUGUE, D., 1975, Sur des tests d’indépendance indépendants de la loi, C.R. Acad. Sci. Paris, t. 281, Ser. A, p. 1103–4.MathSciNetzbMATHGoogle Scholar
  8. (8).
    DVORETZKY, A., KIEFER, J., WOLFOWITZ, J., 1956, Asymptotic minimax character of the classical multinomial operator, Ann. Math. Statist., 27, p. 642–669.MathSciNetCrossRefzbMATHGoogle Scholar
  9. (9).
    EVERITT, B.S., 1977, The analysis of contingency tables, Chapman & Hall.Google Scholar
  10. (10).
    HAJEK, J., SIDAK, Z., 1967, Theory of rank tests, Academic Press.Google Scholar
  11. (11).
    JAMES, B.R., 1975, A functional law of the iterated logarithm for weighted empirical distributions, Ann. Prob., 3, p. 762–772.MathSciNetCrossRefzbMATHGoogle Scholar
  12. (12).
    KENDALL, M., 1970, Rank correlation methods, 4th Edit., Griffin.Google Scholar
  13. (13).
    KIEFER, J., 1961, On large deviations of the empiric D.F. of vector chance variables and a law of the iterated logarithm, Pacific J. Math., 11, p. 649–660.MathSciNetCrossRefzbMATHGoogle Scholar
  14. (14).
    KONIJN, H.S., 1956, On the power of certain tests of independence in bivariate populations, Ann. Math. Statist., 27, p. 300–323.MathSciNetCrossRefzbMATHGoogle Scholar
  15. (15).
    KOZIOL, J.A., NEMEC, A., 1979, On a Cramer-von Mises type statistic for testing bivariate independence, La Revue Canadienne de Statistique, Vol. 7, no 1, p. 43–52.MathSciNetCrossRefzbMATHGoogle Scholar
  16. (16).
    PURI, M.L., SEN, P.K., 1970, Non parametric methods in multivariate analysis, Wiley.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Paul Deheuvels
    • 1
    • 2
  1. 1.Université Paris VI & E.P.H.E.France
  2. 2.Bourg-La-ReineFrance

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