A Ball Divergence Based Measure For Conditional Independence Testing

TL;DR

This paper introduces a new measure for conditional independence based on ball divergence, along with consistent estimators and tests that control error rates, applicable even under complex distributional scenarios.

Contribution

The paper develops a novel conditional dependence measure using ball divergence, with estimators and tests that do not rely on moment assumptions and work under various distributional conditions.

Findings

The proposed tests control Type I error in finite samples.

The tests are asymptotically consistent under various conditions.

The method demonstrates improved power in simulations and real data.

Abstract

In this paper we introduce a new measure of conditional dependence between two random vectors ${\boldsymbol X}$ and ${\boldsymbol Y}$ given another random vector $\boldsymbol Z$ 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-${\boldsymbol X}$ framework and the other based on a novel local wild bootstrap algorithm. In the model-${\boldsymbol X}$ framework, which assumes the knowledge of the distribution of ${\boldsymbol X}|{\boldsymbol Z}$, 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 ${\boldsymbol X}|{\boldsymbol Z}$ is incorrectly specified up to distance preserving transformations. More generally, in situations where ${\boldsymbol X}|{\boldsymbol Z}$ 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.