Simultaneous confidence bands for derivatives of dependent functional data

In this work, consistent estimators and simultaneous confidence bands for the derivatives of mean functions are proposed when curves are repeatedly recorded for each subject. The within-curve correlation of trajectories has been considered while the proposed novel confidence bands still enjoys semiparametric efficiency. The proposed methods lead to a straightforward extension of the two-sample case in which we compare the derivatives of mean functions from two populations. We demonstrate in simulations that the proposed confidence bands are superior to existing approaches which ignore the within-curve dependence. The proposed methods are applied to investigate the derivatives of mortality rates from period lifetables that are repeatedly collected over many years for various countries.