# Scalar-on-function local linear regression and beyond

## TL;DR

This paper advances nonparametric scalar-on-function regression by developing a local linear approach that outperforms local constant methods and enables consistent estimation of functional derivatives, with demonstrated effectiveness on simulated and real data.

## Contribution

It introduces a local linear regression method for scalar-on-function problems using a projection approach, improving estimation accuracy and enabling derivative estimation.

## Key findings

- Local linear regression outperforms local constant methods.
- The estimator of the functional derivative is consistent.
- Good finite sample properties demonstrated on simulated and real data.

## Abstract

Regressing a scalar response on a random function is nowadays a common situation. In the nonparametric setting, this paper paves the way for making the local linear regression based on a projection approach a prominent method for solving this regression problem. Our asymptotic results demonstrate that the functional local linear regression outperforms its functional local constant counterpart. Beyond the estimation of the regression operator itself, the local linear regression is also a useful tool for predicting the functional derivative of the regression operator, a promising mathematical object on its own. The local linear estimator of the functional derivative is shown to be consistent. On simulated datasets we illustrate good finite sample properties of both proposed methods. On a real data example of a single-functional index model we indicate how the functional derivative of the regression operator provides an original and fast, widely applicable estimating method.