BET on Independence
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We study the problem of nonparametric dependence detection. Many existing methods suffer severe power loss due to non-uniform consistency, which we illustrate with a paradox. To avoid such power loss, we approach the nonparametric test of independence through the new framework of binary expansion statistics (BEStat) and binary expansion testing (BET), which examine dependence through a novel binary expansion filtration approximation of the copula. Through a Hadamard-Walsh transform, we find that the cross interactions of binary variables in the filtration are complete sufficient statistics for dependence. These interactions are also uncorrelated under the null. By utilizing these interactions, the BET avoids the problem of non-uniform consistency and improves upon a wide class of commonly used methods (a) by achieving the minimax rate in sample size requirement for specified power and (b) by providing clear interpretations of global and local relationships upon rejection of independence. The binary expansion approach also connects the test statistics with the current computing system to facilitate efficient bitwise implementation. We illustrate the BET by a study of the distribution of stars in the night sky and by an exploratory data analysis of the TCGA breast cancer data.
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