Statistical inference and goodness-of-fit test in functional data via error distribution function

A kernel distribution estimator (KDE) is proposed for the error distribution in the functional data, which is computed from the residuals of the B-spline trajectories over all the measurements. The maximal stochastic process between the KDE and the error distribution is shown to converge to a Gaussian process with mean zero and specified covariance function under some mild conditions. Thus, a simultaneous confidence band (SCB) is constructed for the error distribution based on the KDE in the dense functional data. The proposed SCB is applicable in not only the independent functional data but also the functional time series. In addition, the symmetric test is proposed for the error distribution, as well as a goodness-of-fit test for mean function by using the bootstrap method. Simulation studies examine the finite sample performance of the SCB and show the bootstrap method performs well in numerical studies. The proposed theory is illustrated by the electroencephalogram (EEG) functional data.