The effect of correlated errors on the performance of local linear estimation of regression function based on random functional design

This article considers the problem of nonparametric estimation of the regression function \(r\) in a functional regression model \(Y = r(X) +\varepsilon \) with a scalar response Y, a functional explanatory variable X, and a second order stationary error process \(\varepsilon \). Under some specific criteria, we construct a local linear kernel estimator of \(r\) from functional random design with correlated errors. The exact rates of convergence of mean squared error of the constructed estimator are established for both short and long range dependent error processes. Simulation studies are conducted on the performance of the proposed simple local linear estimator. Examples of time series data are considered.