Distance covariance in metric spaces

Abstract

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Székely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.

Published at http://dx.doi.org/10.1214/12-AOP803 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). This version has appended the two published errata as well

References in corpus (5)

• Measuring and testing dependence by correlation of distances
• Brownian distance covariance
• Discussion of: Brownian distance covariance
• Positive definite metric spaces
• L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry

Cited by in corpus (28)

• Equivalence of distance-based and RKHS-based statistics in hypothesis testing
• A consistent test of independence based on a sign covariance related to Kendall's tau
• Large-Scale Kernel Methods for Independence Testing
• The Chi-Square Test of Distance Correlation
• From Distance Correlation to Multiscale Graph Correlation
• Glucodensities: a new representation of glucose profiles using distributional data analysis
• Asymptotic Distributions of High-Dimensional Distance Correlation Inference
• Discovering and Deciphering Relationships Across Disparate Data Modalities
• The Exact Equivalence of Distance and Kernel Methods for Hypothesis Testing
• The DD${}^G$-classifier in the functional setting
• The Perfect Marriage and Much More: Combining Dimension Reduction, Distance Measures and Covariance
• Estimating Feature-Label Dependence Using Gini Distance Statistics
• Detecting independence of random vectors: generalized distance covariance and Gaussian covariance
• Kernel k-Groups via Hartigan's Method
• Distance Correlation Coefficients for Lancaster Distributions
• Generalized Similarity U: A Non-parametric Test of Association Based on Similarity
• USuRPER: Unit-Sphere Representation PERiodogram for full spectra
• Change Point Methods on a Sequence of Graphs
• Universally Consistent K-Sample Tests via Dependence Measures
• Strong Negative Type in Spheres
• Fourier-type monitoring procedures for strict stationarity
• Using Kernel-Based Statistical Distance to Study the Dynamics of Charged Particle Beams in Particle-Based Simulation Codes
• Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance
• Sphere Theorems with and without Smoothing
• Adaptation of the Phase Distance Correlation Periodogram to Account for Measurement Uncertainties
• Change-Point Detection for Object-valued Time Series
• Model Independent Periodogram for Scanning Astrometry
• Interpretation and visualization of distance covariance through additive decomposition of correlations formula