Strong consistency of the local linear estimator for a generalized regression function with dependent functional data

TL;DR

This paper establishes the strong consistency and convergence rates of a local linear estimator for a generalized regression model with dependent functional data, demonstrating its superior performance in simulations and energy forecasting.

Contribution

It provides the first almost complete convergence rates for the local linear estimator in a dependent functional data setting, extending previous results to heterogeneously distributed and strongly mixing data.

Findings

The local linear estimator achieves the same convergence rates pointwise and uniformly on compact sets.

Dependent data can slow convergence rates compared to independent data.

The functional local linear estimator outperforms the local constant estimator in simulations and energy forecasting.

Abstract

In this study, we focus on a generalized nonparametric scalar-on-function regression model for heterogeneously distributed and strongly mixing data. We provide almost complete convergence rates for the local linear estimator of the regression function. We show that, under our conditions, the pointwise and uniform convergence rates are the same on a compact set. On the other hand, when the data is dependent, it is proved that the convergence rate can be slower than those obtained for independent data. A simulation study shows the good performance and finite sample properties of the functional local linear estimator (FLL) in comparison to the local constant estimator (FLC). In addition, a one step ahead energy consumption forecasting exercise illustrates that the forecasts of the FLL estimator are significantly more accurate than those of the FLC.