Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yin Tang, Bing Li

Abstract

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.

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

Abstract

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.
Paper Structure (26 sections, 10 theorems, 158 equations, 7 figures, 5 tables)

This paper contains 26 sections, 10 theorems, 158 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background of RKHS
  3. Construction of nested Hilbert spaces
  4. Mean element and covariance operator
  5. Conditional Covariance Operator (CCO)
  6. Test Construction for Conditional Independence
  7. Construction of Test Statistic
  8. Estimating CCCO and its Hilbert-Schmidt norm
  9. Asymptotic Distribution of $T _{n}$
  10. Approximation for Random Functions
  11. Approximating trajectories by kernel-based smoother
  12. Approximating inner products and distances
  13. Simplification under balanced observation schedule
  14. Sample-Level Implementation
  15. Estimation of eigenvalues
  16. ...and 11 more sections

Key Result

Lemma 1

If Assumptions ass-moment and ass-cond-exp are satisfied, then, for all $f \in \ker(\Sigma_{ZZ})^\perp \subset \mathcal{G}_Z$, $g \in \mathcal{G}_X$ and $h \in \mathcal{G}_Y$, we have

Figures (7)

  • Figure 1: P-values for simulations under balanced and unbalanced cases for Model 1. The red line represents $p=0.05$.
  • Figure 2: P-values for simulations under balanced and unbalanced cases for Model 2. The red line represents $p=0.05$.
  • Figure 3: P-values for simulations under balanced and unbalanced cases for Model 3. The red line represents $p=0.05$.
  • Figure 4: P-values for simulations under balanced and unbalanced cases for Model 4. The red line represents $p=0.05$.
  • Figure 4: Sample sizes for different tests. The row and column indices 1--7 correspond to the factors in Table \ref{['tab:factors']}.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Definition 1: characteristic kernel
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Corollary 2
  • Corollary 3
  • Theorem 1
  • ...and 7 more