A Ball Divergence Based Measure For Conditional Independence Testing

In this paper we introduce a new measure of conditional dependence between two random vectors X and Y given another random vector Z using the ball divergence. Our measure characterizes conditional independence and does not require any moment assumptions. We propose a consistent estimator of the measure using a kernel averaging technique and derive its asymptotic distribution. Using this statistic we construct two tests for conditional independence, one in the model- X framework and the other based on a novel local wild bootstrap algorithm. In the model- X framework, which assumes the knowledge of the distribution of X| Z, applying the conditional randomization test we obtain a method that controls Type I error in finite samples and is asymptotically consistent, even if the distribution of X| Z is incorrectly specified up to distance preserving transformations. More generally, in situations where X| Z is unknown or hard to estimate, we design a double-bandwidth based local wild bootstrap algorithm that asymptotically controls both Type I error and power. We illustrate the advantage of our method, both in terms of Type I error and power, in a range of simulation settings and also in a real data example. A consequence of our theoretical results is a general framework for studying the asymptotic properties of a 2-sample conditional V-statistic, which is of independent interest.