A Kernel-Based Nonparametric Test for Conditional Independence of Functional Data

Conditional independence is a fundamental concept in many areas of statistical research, including, for example, sufficient dimension reduction, causal inference, and statistical graphical models. In many modern applications, data arise in the form of random functions, making it important to determine whether two random functions are conditionally independent given a third. However, to the best of our knowledge, existing conditional independence tests in the literature apply only to multivariate data, and extensions to the functional setting are not available. To fill this gap, we develop a kernel-based test for conditional independence of random functions based on the conjoined conditional covariance operator (CCCO). We rigorously derive the asymptotic distribution of the CCCO estimator using a recently established sharpened convergence rate for the regression operator (Choi et al., 2026). Based on this result, we construct a test statistic using the spectral decomposition of the operator appearing in the asymptotic distribution. The proposed method is illustrated through applications to an activity and biometrics dataset and a macroeconomic dataset.

💡 Research Summary

This paper addresses the problem of testing conditional independence (CI) among three random elements that are functions rather than finite‑dimensional vectors. While CI testing is a cornerstone of sufficient dimension reduction, causal inference, and graphical modeling, existing CI tests are limited to multivariate data. The authors fill this gap by developing a kernel‑based, fully non‑parametric CI test for functional data, built on the conjoined conditional covariance operator (CCCO) defined in a reproducing kernel Hilbert space (RKHS).

The methodological core begins with a review of RKHS theory: mean elements, covariance operators, and cross‑covariance operators are introduced for random functions X, Y, and Z. Assuming characteristic kernels (e.g., Gaussian or Laplace) ensures that these operators capture the full probability law. The conditional covariance operator is defined as
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