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A nonparametric measure of independence under a hypothesis of independent components. (English) Zbl 0770.62039

It is shown that, under certain regularity assumptions, the weighted \(L^ 2\)-distance \[ I_ n=\iint\bigl[f_ n(x_ 1,x_ 2)-g_ n(x_ 1)h_ n(x_ 2)\bigr]^ 2 a(x_ 1,x_ 2) dx_ 1 dx_ 2 \] between a bivariate kernel density estimator \(f_ n\) and the tensor product of its marginals \(g_ n\) and \(h_ n\) is asymptotically normal after a proper standardization, provided that the components of the observed (bivariate) data are independent.
The method of proof utilizes a central limit theorem for degenerate \(U\)- statistics due to P. Hall [J. Multivariate Anal. 14, 1-16 (1984; Zbl 0528.62028)].
Reviewer: W.Stute (Gießen)

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0528.62028
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References:

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