Distance correlation is a new measure of dependence between random vectors.
Distance covariance and distance correlation are analogous to product-moment
covariance and correlation, but unlike the classical definition of correlation,
distance correlation is zero only if the random vectors are independent. The
empirical distance dependence measures are based on certain Euclidean distances
between sample elements rather than sample moments, yet have a compact
representation analogous to the classical covariance and correlation.
Asymptotic properties and applications in testing independence are discussed.
Implementation of the test and Monte Carlo results are also presented.Comment: Published in at http://dx.doi.org/10.1214/009053607000000505 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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 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.
Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q) = 1. We prove that for an arbitraryA similar statement is not true for the supremum: if 1 < x < 2 and Z 1 , Z 2 are independent standard Gaussian rv's, then sup 0≤λ≤1/2 P {λZ 2 1 + (1 − λ)Z 2 2 ≤ x} is taken not at λ = 0 or at λ = 1/2 but at 0 < λ = λ(x) < 1/2, where λ(x) is a continuous, increasing function from λ(1) = 0 to λ(2) = 1/2, e.g. λ(1.5) = .15 . . .. Applications of our theorems include asymptotic quantiles of U and V -statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes.
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