Four simple axioms of dependence measures

@article{Mri2018FourSA,
  title={Four simple axioms of dependence measures},
  author={Tam{\'a}s F. M{\'o}ri and G{\'a}bor J. Sz{\'e}kely},
  journal={Metrika},
  year={2018},
  volume={82},
  pages={1-16}
}
Recently new methods for measuring and testing dependence have appeared in the literature. One way to evaluate and compare these measures with each other and with classical ones is to consider what are reasonable and natural axioms that should hold for any measure of dependence. We propose four natural axioms for dependence measures and establish which axioms hold or fail to hold for several widely applied methods. All of the proposed axioms are satisfied by distance correlation. We prove that… 
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References

SHOWING 1-10 OF 50 REFERENCES
On Quantifying Dependence: A Framework for Developing Interpretable Measures
We present a framework for selecting and developing measures of dependence when the goal is the quantification of a relationship between two variables, not simply the establishment of its existence.
Measuring Dependence Powerfully and Equitably
TLDR
This paper introduces and characterize a population measure of dependence called MIC*, and introduces an efficient approach for computing MIC* from the density of a pair of random variables, and defines a new consistent estimator MICe for MIC* that is efficiently computable.
Distance Covariance in Metric Spaces: Non-Parametric Independence Testing in Metric Spaces (Master's thesis)
The aim of this thesis is to find a solution to the non-parametric independence problem in separable metric spaces. Suppose we are given finite collection of samples from an i.i.d. sequence of paired
New dependence coefficients. Examples and applications to statistics
Abstract.To measure the dependence between a real-valued random variable X and a σ-algebra , we consider four distances between the conditional distribution function of X given and the distribution
Some Concepts of Dependence
Problems involving dependent pairs of variables (X, Y) have been studied most intensively in the case of bivariate normal distributions and of 2 × 2 tables. This is due primarily to the importance of
A non-parametric test of independence ∗
We propose a new class of nonparametric tests for the supposition of independence between two continuous random variables X and Y. Given a sample of (X,Y ), the tests are based on the size of the
Measuring and testing dependence by correlation of distances
TLDR
Distance correlation is a new measure of dependence between random vectors that is based on certain Euclidean distances between sample elements rather than sample moments, yet has a compact representation analogous to the classical covariance and correlation.
Equivalence of distance-based and RKHS-based statistics in hypothesis testing
TLDR
It is shown that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.
A Non-Parametric Test of Independence
A test is proposed for the independence of two random variables with continuous distribution function (d.f.). The test is consistent with respect to the class Ω′of d.f.’s with continuous joint and
Distance Correlation: A New Tool for Detecting Association and Measuring Correlation Between Data Sets
The difficulties of detecting association, measuring correlation, and establishing cause and effect have fascinated mankind since time immemorial. Democritus, the Greek philosopher, underscored well
...
...