Oracle-efficient estimation for functional data error distribution with simultaneous confidence band

Abstract

Kolmogorov-Smirnov (K-S) simultaneous confidence band (SCB) is constructed for the error distribution of dense functional data based on kernel distribution estimator (KDE). The KDE is computed from residuals of B spline trajectories over a smaller number of measurements, whereas the B spline trajectories are computed from the remaining larger set of measurements. Under mild and simple assumptions, it is shown that the KDE is a uniformly oracle-efficient estimator of the error distribution, and the SCB has the same asymptotic properties as the classic K-S SCB based on the infeasible empirical cumulative distribution function (EDF) of unobserved errors. Simulation examples corroborate with the theoretical findings. The proposed method is illustrated by examples of an EEG (Electroencephalogram) data and a stock data.