Bootstrap-based inference on the difference in the means of two correlated functional processes

We propose nonparametric inference methods on the mean difference between two correlated functional processes. We compare methods that (1) incorporate different levels of smoothing of the mean and covariance; (2) preserve the sampling design; and (3) use parametric and nonparametric estimation of the mean functions. We apply our method to estimating the mean difference between average normalized δ power of sleep electroencephalograms for 51 subjects with severe sleep apnea and 51 matched controls in the first 4;h after sleep onset. We obtain data from the Sleep Heart Health Study, the largest community cohort study of sleep. Although methods are applied to a single case study, they can be applied to a large number of studies that have correlated functional data.