A Note on Local Polynomial Regression for Time Series in Banach Spaces

ABSTRACT This work extends local polynomial regression to Banach space‐valued time series for estimating smoothly varying means and their derivatives in non‐stationary data. The asymptotic properties of both the standard and bias‐reduced Jackknife estimators are analyzed under mild moment conditions, establishing their convergence rates. Simulation studies assess the finite sample performance of these estimators and compare them with the Nadaraya–Watson estimator. Additionally, the proposed methods are applied to smooth EEG recordings for reconstructing eye movements and to video analysis for detecting pedestrians and abandoned objects.