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Abstract:
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This paper gives a state-of-the-art comparison of some new measures of dependence. Most specifically, this paper identifies which of the seven postulates of Alfred Renyi (1959) about the measures of dependence can be established or disproved for each of the measures. The measures that brought interest to the statistical community are distance correlation (dCor) by Szekely and Rizzo (2007), maximal information coefficient (MIC) by Reshef, Reshef, Finucane, Grossman, McVean, Turnbaugh and Sabeti (2011), and global Gaussian correlation (GGC) by Berentsen and Tjostheim (2014). We have observed that none of Renyi's postulates were satisfied by MIC since Reshef et al. did not define a population counterpart for it. GGC, as well as dCor, satisfies three postulates of Renyi, two of which are common to both. These properties are that of symmetry and being a number between 0 and 1. Global Gaussian correlation has the property that it is equal to the absolute value of the Pearson product moment correlation. However, dCor is the only one that has the important characterization of being equal to zero if and only if two random variables are independent.
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