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A multivariate nonparametric test of independence. (English) Zbl 1099.62042

Summary: A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of the product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.

MSC:

62G10 Nonparametric hypothesis testing
62H20 Measures of association (correlation, canonical correlation, etc.)
62E20 Asymptotic distribution theory in statistics
62H15 Hypothesis testing in multivariate analysis
65C05 Monte Carlo methods
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