Non parametric tests of independence
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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  and .
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 , for n=11−30, extending the tabulation for n=3−10 obtained in .
Key wordsNon 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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