## Abstract

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 if a dependence measure is defined for all bounded nonconstant real valued random variables and is invariant with respect to all one-to-one measurable transformations of the real line, then the dependence measure cannot be weakly continuous. This implies that the classical maximal correlation cannot be continuous and thus its application is problematic. The recently introduced maximal information coefficient has the same disadvantage. The lack of weak continuity means that as the sample size increases the empirical values of a dependence measure do not necessarily converge to the population value.

## Keywords

Correlation Distance correlation Maximal correlation Maximal information coefficient Invariance## References

- Dedecker J, Prieur C (2005) New dependence coefficients. Examples and applications to statistics. Probab Theory Relat Fields 132:203–236. https://doi.org/10.1007/s00440-004-0394-3 MathSciNetCrossRefzbMATHGoogle Scholar
- Dueck J, Edelmann D, Gneiting T, Richards D (2014) The affinely invariant distance correlation. Bernoulli 20:2305–2330. https://doi.org/10.3150/13-BEJ558 MathSciNetCrossRefzbMATHGoogle Scholar
- Eaton ML (1989) Group invariance. Applications in statistics, NSF-CBMS regional conference series in probability and statistics 1. IMS, HaywardGoogle Scholar
- Escoufier Y (1973) Le Traitement des Variables Vectorielles. Biometrics 29:751–760. https://doi.org/10.2307/2529140 MathSciNetCrossRefGoogle Scholar
- Gebelein H (1941) Das statistische Problem der Korrelation als Variations- und Eigenwert-problem und sein Zusammenhang mit der Ausgleichungsrechnung. Z Angew Math Mech 21:364–379. https://doi.org/10.1002/zamm.19410210604 MathSciNetCrossRefzbMATHGoogle Scholar
- Gouvêa FQ (2011) Was cantor surprised? Am Math Mon 118:198–209. https://doi.org/10.4169/amer.math.monthly.118.03.198 MathSciNetCrossRefzbMATHGoogle Scholar
- Hirschfeld HO (1935) A connection between correlation and contingency. Math Proc Camb Philos Soc 31:520–524. https://doi.org/10.1017/S0305004100013517 CrossRefzbMATHGoogle Scholar
- Hoeffding W (1940) Masstabinvariante Korrelationstherie. Schr Math Inst und Inst Angew Math Univ Berlin 5:181–233Google Scholar
- Hoeffding W (1948) A non-parametric test of independence. Ann Math Stat 19:546–557. https://doi.org/10.1214/aoms/1177730150 MathSciNetCrossRefzbMATHGoogle Scholar
- Huang Q, Zhu Y (2016) Model-free sure screening via maximum correlation. J Multivar Anal 148:89–106. https://doi.org/10.1016/j.jmva.2016.02.014 MathSciNetCrossRefzbMATHGoogle Scholar
- Jakobsen ME (2017) Distance covariance in metric spaces: non-parametric independence testing in metric spaces. arXiv:1706.03490. Accessed 9 Jan 2018
- Josse J, Holmes S (2014) Tests of independence and beyond. arXiv:1307.7383v3. Accessed 9 Jan 2018
- Kendall MG (1938) A new measure of rank correlation. Biometrika 30:81–93. https://doi.org/10.2307/2332226 CrossRefzbMATHGoogle Scholar
- Kimeldorf G, Sampson AR (1978) Monotone dependence. Ann Stat 6:895–903. https://doi.org/10.1214/aos/1176344262 MathSciNetCrossRefzbMATHGoogle Scholar
- Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37:1137–1153. https://doi.org/10.1214/aoms/1177699260 MathSciNetCrossRefzbMATHGoogle Scholar
- Lehmann EL, Romano JP (2005) Testing statistical hypotheses, 3rd edn. Springer, New York. https://doi.org/10.1007/0-387-27605-X zbMATHGoogle Scholar
- Linfoot EH (1957) An informational measure of correlation. Inf Control 1:85–89. https://doi.org/10.1016/S0019-9958(57)90116-X MathSciNetCrossRefzbMATHGoogle Scholar
- López Blázquez F, Salamanca Miño B (2014) Maximal correlation in a non-diagonal case. J Multivar Anal 131:265–278. https://doi.org/10.1016/j.jmva.2014.07.008 MathSciNetCrossRefzbMATHGoogle Scholar
- Lyons R (2013) Distance covariance in metric spaces. Ann Probab 41:3284–3305. https://doi.org/10.1214/12-AOP803 MathSciNetCrossRefzbMATHGoogle Scholar
- Papadatos N (2014) Some counterexamples concerning maximal correlation and linear regression. J Multivar Anal 126:114–117. https://doi.org/10.1016/j.jmva.2013.12.008 MathSciNetCrossRefzbMATHGoogle Scholar
- Papadatos N, Xifara T (2013) A simple method for obtaining the maximal correlation coefficient and related characterizations. J Multivar Anal 118:102–114. https://doi.org/10.1016/j.jmva.2013.03.017 MathSciNetCrossRefzbMATHGoogle Scholar
- Pearson K (1920) Notes on the history of correlation. Biometrika 13:25–45. https://doi.org/10.2307/2331722 CrossRefGoogle Scholar
- Reimherr M, Nicolae DL (2013) On quantifying dependence: a framework for developing interpretable measures. Stat Sci 28:116–130. https://doi.org/10.1214/12-STS405 MathSciNetCrossRefzbMATHGoogle Scholar
- Rényi A (1959) On measures of dependence. Acta Mat Acad Sci Hung 10:441–451. https://doi.org/10.1007/BF02024507 MathSciNetCrossRefzbMATHGoogle Scholar
- Reshef DN, Reshef YA, Finucane HK, Grossman SR, McVean G, Turnbaugh PJ, Lander ES, Mitzenmacher M, Sabeti PC (2011) Detecting novel associations in large data sets. Science 334(6062):1518–1524. https://doi.org/10.1126/science.1205438 CrossRefzbMATHGoogle Scholar
- Reshef YA, Reshef DN, Finucane HK, Sabeti PC, Mitzenmacher M (2016) Measuring dependence powerfully and equitably. J Mach Learn Res 17(212):1–63MathSciNetzbMATHGoogle Scholar
- Richards DStP (2017) Distance correlation: a new tool for detecting association and measuring correlation between data sets. Plenary talk at the Joint Mathematics Meeting, Atlanta, 2017. Not Am Math Soc 64:16–18. https://doi.org/10.1090/noti1457 Google Scholar
- Sampson AR (1984) A multivariate correlation ratio. Stat Probab Lett 2:77–81. https://doi.org/10.1016/0167-7152(84)90054-3 MathSciNetCrossRefzbMATHGoogle Scholar
- Schweizer B, Wolff EF (1981) On nonparametric measures of dependence for random variables. Ann Stat 9:879–885. https://doi.org/10.1214/aos/1176345528 MathSciNetCrossRefzbMATHGoogle Scholar
- Sejdinovic D, Sriperumbudur B, Gretton A, Fukumiyu K (2013) Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Ann Stat 41:2263–2291. https://doi.org/10.1214/13-AOS1140 MathSciNetCrossRefzbMATHGoogle Scholar
- Simon N, Tibshirani R (2011) Comment on “Detecting novel associations in large data set” by Reshef et al. Science Dec 16, 2011. arXiv:1401.7645v1. Accessed 9 Jan 2018
- Spearman C (1904) A proof and measurement of association between two things. Am J Psychol 15:72–101. https://doi.org/10.2307/1412159 CrossRefGoogle Scholar
- Speed T (2011) A correlation for the 21st century. Science 334(6062):1502–1503. https://doi.org/10.1126/science.1215894 CrossRefGoogle Scholar
- Stigler S (1989) Francis Galton’s account of the invention of correlation. Stat Sci 4:73–79. https://doi.org/10.1214/ss/1177012580 MathSciNetCrossRefzbMATHGoogle Scholar
- Székely GJ, Rizzo ML, Bakirov NK (2007) Measuring and testing independence by correlation of distances. Ann Stat 35:2769–2794. https://doi.org/10.1214/009053607000000505 CrossRefzbMATHGoogle Scholar
- Székely GJ, Rizzo ML (2009) Brownian distance covariance. Ann Appl Stat 3:1236–1265. https://doi.org/10.1214/09-AOAS312 MathSciNetCrossRefzbMATHGoogle Scholar
- Székely GJ, Rizzo ML (2014) Partial distance correlation with methods for dissimilarities. Ann Stat 42:2382–2412. https://doi.org/10.1214/14-AOS1255 MathSciNetCrossRefzbMATHGoogle Scholar
- Volokh E (2015) Zero correlation between State Homicide and State Gun Laws. The Washington Post, October 6, 2015Google Scholar