Bootstrapping sums of independent but not identically distributed continuous processes with applications to functional data

In many areas of application, the data are of functional nature, such as (one-dimensional) spectral data and two- or three-dimensional imaging data. It is often of interest to test for the significance of some set of factors in the functional observations (e.g., test for the mean differences between two groups). Testing hypotheses point-by-point (voxel-by-voxel in neuroimaging studies) results in a severe multiple-comparisons problem as the number of measurements made per observation is typically much larger than the number of observations (“large pp, small nn”). Thus solutions to this problem should take into account the spatial correlation structure inherent in the data. Popular approaches in such a setting include the general Statistical Parametric Mapping (SPM) approach and the permutation test, but these rely on strong parametric and exchangeability assumptions. In situations in which these assumptions are not satisfied, a nonparametric multiplier bootstrap approach may be used. Motivated by this problem, we present general results for multiplier bootstraps for sums of independent but not identically distributed processes. We also consider the application of these results to an imaging setting and provide sufficient conditions that will ensure asymptotic control of the familywise error rate.