Functional delta residuals and applications to functional effect sizes

Given a functional central limit (fCLT) and a parameter transformation, we use the functional delta method to construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the transformed limit process. Moreover, we prove a multiplier bootstrap fCLT theorem for these transformed residuals and show how this can be used to construct simultaneous confidence bands for transformed functional parameters. As motivation for this methodology, we provide the formal application of these residuals to a functional version of the effect size parameter Cohen's $d$, a problem appearing in current brain imaging applications. The performance and necessity of such residuals is illustrated in a simulation experiment for the covering rate of simultaneous confidence bands for the functional Cohen's $d$ parameter.