# Measuring and testing dependence by correlation of distances

@article{Szekely2007MeasuringAT, title={Measuring and testing dependence by correlation of distances}, author={G'abor J. Sz'ekely and Maria L. Rizzo and Nail K. Bakirov}, journal={Annals of Statistics}, year={2007}, volume={35}, pages={2769-2794} }

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… Expand

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#### References

SHOWING 1-10 OF 19 REFERENCES

A multivariate nonparametric test of independence

- Mathematics
- 2006

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… Expand

A new test for multivariate normality

- Mathematics
- 2005

We propose a new class of rotation invariant and consistent goodness-of-fit tests for multivariate distributions based on Euclidean distance between sample elements. The proposed test applies to any… Expand

Hierarchical Clustering via Joint Between-Within Distances: Extending Ward's Minimum Variance Method

- Mathematics, Computer Science
- J. Classif.
- 2005

A hierarchical clustering method that minimizes a joint between-within measure of distance between clusters, by defining a cluster distance and objective function in terms of Euclidean distance, or any power of Euclidesan distance in the interval (0,2). Expand

Correlational Meta-Analysis: Independent and Nonindependent Cases

- Mathematics
- 1992

The purpose of this study was to determine the effect of the violation of the assumption of independence when combining correlation coefficients in a meta-analysis. In this Monte Carlo simulation the… Expand

On the Independence of k Sets of Normally Distributed Statistical Variables

- Mathematics
- 1935

IN SUCH fields of investigation as economics, psychology, and anthropology, where observations on several variables are taken into account simultaneously, it is at least as important to study… Expand

DISTRIBUTION-FREE AND ROBUST STATISTICAL METHODS: VIABLE ALTERNATIVES TO PARAMETRIC STATISTICS?

- Biology
- 1993

This paper attempts to introduce some distribution-free and robust techniques to ecologists and to offer a critical appraisal of the potential advantages and drawbacks of these methods. Expand

Extremal probabilities for Gaussian quadratic forms

- Mathematics
- 2003

Abstract. Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, infQP(Q≤x)=P(χ2n/n≤x), where χn2 is… Expand

Group Invariance in Statistical Inference

- Mathematics
- 1996

Group invariance matrices, groups and Jacobians invariance equivariant estimation in curved models some best invarient tests in multinormals minimax tests in multinormals locally minimax tests in… Expand

Limitations of the case-only design for identifying gene-environment interactions.

- Medicine
- American journal of epidemiology
- 2001

The authors explore the robustness of this procedure to uncertainty about the independence assumption by using simulations and demonstrate that inferences about the multiplicative interaction with the case-only design can be highly distorted when there is departure from the independent assumption. Expand

Multivariate Analysis

- Nature
- 1958

An Introduction to Multivariate Statistical AnalysisBy Prof. T. W. Anderson. (Wiley Publications in Mathematical Statistics.) Pp. xii + 374. (New York: John Wiley and Sons, Inc.; London: Chapman and… Expand