Conditional independence testing in Hilbert spaces with applications to functional data analysis

We 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 variables. This generalises, for example, testing the significance of X in a scalar-on-function linear regression model of response Y on functional regressors X and Z. We show however that even in the idealised setting where additionally (X,Y, Z) have a non-singular Gaussian distribution, the power of any test cannot exceed its size. Further modelling assumptions are needed to restrict the null and we argue that a convenient way of specifying these is based on choosing methods for regressing each of X and Y on Z. We thus propose as a test statistic, the Hilbert–Schmidt norm of the outer product of the resulting residuals, and prove that type I error control is guaranteed when the in-sample prediction errors are sufficiently small. We show this requirement is met by ridge regression in functional linear model settings without requiring any eigen-spacing conditions or lower bounds on the eigenvalues of the covariance of the functional regressor. We apply our test in constructing confidence intervals for truncation points in truncated functional linear models.