Research Projects

Testing independence and conditional independence among random variables is fundamental to statistical inference, with broad applications in areas such as feature selection, independent component analysis, causal inference, and graphical model learning. While a wide range of methods exists for testing dependence among univariate random variables, there remain gaps in the literature - particularly in testing joint independence among more than two multivariate random vectors. Research gaps also exist in the domain of conditional independence testing as many existing methods struggle with the curse of dimensionality, sensitivity to sample size, and lack of robustness. This project builds upon the work of Roy et al. (2020), who proposed a novel framework that constructs a dependence measure for random vectors based on a univariate dependence measure. However, several important theoretical properties of this approach remain unaddressed in their work --- such as omnibus consistency, asymptotic distribution, local power, and the behavior of the test under transformations. Moreover, the approach has not been extended to test for conditional independence. Our research plan proceeds in two parts. The first part focuses on deepening the theoretical understanding of the original framework of Roy et al. (2020). We aim to establish the omnibus consistency and asymptotic distribution of the estimator. We also plan to characterize conditions under which the test statistic achieves its maximum and to analyze its local power. A comprehensive numerical comparison will be conducted using different choices of univariate dependence measures to evaluate the strengths and limitations of the method in diverse scenarios. In the second part, we aim to develop a new conditional independence test. Inspired by the conditional distance covariance approach of Wang et al. (2015), we define a conditional version based on the framework proposed by Roy et al. (2020). We investigate its theoretical properties, construct a consistent estimator, and derive its asymptotic distribution. Finally, we will evaluate the empirical performance of the proposed test through simulation studies and applications to real-world datasets. We believe this project will make substantial contributions --- both theoretical and methodological --- to the literature on testing independence and conditional independence.