Testing relevant hypotheses in functional variance function via self‐normalization

We propose a novel methodology for testing relevant hypotheses in the functional variance functions of contaminated functional data via spline‐backfitted kernel smoothing and self‐normalization. Our approach focuses on testing the null hypothesis of no relevant deviation instead of exact equality, such as the equality of two variance functions from two independent measurement errors. The proposed statistics enable testing of relevant hypotheses in one‐sample, two‐sample, and single or multiple change points problems, and exhibit oracle efficiency, meaning that developed procedures are asymptotically indistinguishable from those with true trajectories. Additionally, we demonstrate the finite sample properties of our proposed tests using a simulation study and electroencephalogram (EEG) data.