Gábor J. Székely

Measuring and testing dependence by correlation of distances

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.

BROWNIAN DISTANCE COVARIANCE

We discuss briefly the very interesting concept of Brownian distance covariance developed by Székely and Rizzo (2009) and describe two possible extensions. The first extension is for high dimensional data that can be coerced into a Hilbert space, including certain high throughput screening and functional data settings. The second extension involves very simple modifications that may yield increased power in some settings. We commend Székely and Rizzo for their very interesting work and recognize that this general idea has potential to have a large impact on the way in which statisticians evaluate dependency in data.Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. Distance correlation characterizes independence: it is zero if and only if the random vectors are independent. The notion of covariance with respect to a stochastic process is introduced, and it is shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product-moment covariance, as we obtain Pearson product-moment covariance by replacing the Brownian motion in the defin- ition with identity. The corresponding statistic has an elegantly simple com- puting formula. Advantages of applying Brownian covariance and correlation vs the classical Pearson covariance and correlation are discussed and illustrated.

The distance correlation t-test of independence in high dimension

Distance correlation is extended to the problem of testing the independence of random vectors in high dimension. Distance correlation characterizes independence and determines a test of multivariate independence for random vectors in arbitrary dimension. In this work, a modified distance correlation statistic is proposed, such that under independence the distribution of a transformation of the statistic converges to Student t, as dimension tends to infinity. Thus we obtain a distance correlation t-test for independence of random vectors in arbitrarily high dimension, applicable under standard conditions on the coordinates that ensure the validity of certain limit theorems. This new test is based on an unbiased estimator of distance covariance, and the resulting t-test is unbiased for every sample size greater than three and all significance levels. The transformed statistic is approximately normal under independence for sample size greater than nine, providing an informative sample coefficient that is easily interpretable for high dimensional data.

DISCO analysis: A nonparametric extension of analysis of variance

In classical analysis of variance, dispersion is measured by considering squared distances of sample elements from the sample mean. We consider a measure of dispersion for univariate or multivariate response based on all pairwise distances between-sample elements, and derive an analogous distance components (DISCO) decomposition for powers of distance in (0, 2]. The ANOVA F statistic is obtained when the index (exponent) is 2. For each index in (0, 2), this decomposition determines a nonparametric test for the multi-sample hypothesis of equal distributions that is statistically consistent against general alternatives.

Partial distance correlation with methods for dissimilarities

Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions, and applications of distance correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation we introduce a new Hilbert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hilbert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.

On multivariate skewness and kurtosis

Let X be a d-dimensional standardized random variable

A note on the background of several Bonferroni−Galambos-type inequalities

({\bf E}(X) = 0, \operatorname{cov} (X) = 1)

Sharp inequalities between skewness and kurtosis

. Then for a multivariate analogue of skewness

An extremal property of rectangular distributions

s = {\bf E}(\| X \|^2 X)

Fast Computing for Distance Covariance

and kurtosis