Single index regression for locally stationary functional time series

In this research, we formulated an asymptotic theory for single index regression applied to locally stationary functional time series. Our approach involved introducing estimators featuring a regression function that exhibited smooth temporal changes. We rigorously established the uniform convergence rates for kernel estimators, specifically the Nadaraya-Watson (NW) estimator for the regression function. Additionally, we provided a central limit theorem for the NW estimator. Finally, the theory was supported by a comprehensive simulation study to investigate the finite-sample performance of our proposed method.