Rank-based Coefficients for (conditional) Independence Testing
YUHAO WANG – TSINGHUA UNIVERSITY
ABSTRACT
This talk presents two rank-based coefficients for independence and conditional independence testing. In the first part, we introduce a multivariate extension of the Azadkia-Chatterjee rank coefficient, extending the original measure of dependence between a random variable and a random vector to two random vectors Y and Z. Based on ni.i.d. observations, the proposed coefficient converges almost surely to a population quantity that lies in [0, 1], equals zero if and only if Y and Z are independent, and equals one if and only if Y is almost surely a function of Z. These properties require only that Y is not almost surely a constant vector. Under the same mild condition, we further establish asymptotic normality under independence, which leads to a Wald-type test.
In the second part, we focus directly on testing the conditional independence of Y and Z, conditional on a third random variable X. Existing approaches often rely on measures of conditional correlation and may therefore have limited sensitivity to highly nonlinear forms of dependence. We propose a rank-based coefficients that has two properties. First, it can universally reject all alternatives. Second, it satisfies a “smoothness doubly robust” property: its asymptotic validity requires only one of the conditional CDFs P(Y <= y | X=x) and P(Z <= z | X=x) to be sufficiently smooth in x, while the other may be arbitrarily complex, without requiring prior knowledge of which one is smooth.