Title: Conditional independence testing for functional data
Authors: Anton Rask Lundborg - University of Cambridge (United Kingdom) [presenting]
Rajen D Shah - University of Cambridge (United Kingdom)
Jonas Peters - University of Copenhagen (Denmark)
Abstract: The aim is to study the problem of testing the null hypothesis that $X$ and $Y$ are conditionally independent given $Z$, where each of $X$, $Y$ and $Z$ may be functional random elements. We show that even in the idealised setting where $(X, Y, Z)$ are jointly Gaussian with Z infinite-dimensional (i.e., functional) any test with power $\beta$ at an alternative, must reject some null with probability at least $\beta$. Given the untestability of this hypothesis, we argue that tests must be designed so their suitability for a particular problem may be judged easily. To this end, we propose regressing each of $X$ and $Y$ onto $Z$ and then computing the Hilbert-Schmidt norm of the outer product of the resulting residuals. We show that the level of the resulting test is controlled uniformly over a class of distributions governed primarily by the requirement that the regressions estimate the conditional expectations of $X$ and $Y$ given $Z$ sufficiently well. Whilst our result allows for arbitrary regression methods, we develop the theoretical guarantees for Tikhonov regularised regressions. Simulations studies demonstrate the effectiveness of our approach for conditional independence testing and its applications in variable selection and truncation point estimation in functional linear models.