Expected Conditional Characteristic Function-based Measures for Testing Independence

Abstract

We propose a novel class of independence measures for testing independence between two random vectors based on the discrepancy between the conditional and the marginal characteristic functions. The relation between our index and other similar measures is studied, which indicates that they all belong to a large framework of reproducing kernel Hilbert space. If one of the variables is categorical, our asymmetric index extends the typical ANOVA to a kernel ANOVA that can test a more general hypothesis of equal distributions among groups. In addition, our index is also applicable when both variables are continuous. We develop two empirical estimates and obtain their respective asymptotic distributions. We illustrate the advantages of our approach by numerical studies across a variety of settings including a real data example. Supplementary materials for this article are available online.

Keywords:

• Categorical variables
• Characteristic function
• Independence
• Reproducing kernel Hilbert space

Acknowledgments

The authors thank the editor, the associate editor, and four referees for their insightful and constructive comments, which lead to a greatly improved version. This work is supported in part by an NSF grant CIF-1813330.