Abstract
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 (j)n , 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 (1)n (x1),...,F (p)n (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].
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Deheuvels, P. (1980). Non parametric tests of independence. In: Raoult, JP. (eds) Statistique non Paramétrique Asymptotique. Lecture Notes in Mathematics, vol 821. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097426
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DOI: https://doi.org/10.1007/BFb0097426
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