  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 testing the significance of X in a
regression model of scalar response Y on functional regressors X and Z. We show
however that even in the idealised setting where additionally (X, Y, Z) has a
Gaussian distribution, the power of any test cannot exceed its size. Further
modelling assumptions are needed 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 propose a test statistic involving inner products of the resulting
residuals that is simple to compute and calibrate: type I error is controlled
uniformly 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 and testing for edges in a functional graphical model for EEG
data.
