# On estimation of the effect lag of predictors and prediction in functional linear model

## TL;DR

This paper introduces a functional linear model that predicts responses using multiple functional predictors, estimates their effect lags, and evaluates the model's properties and performance through simulations.

## Contribution

It presents a novel method for estimating predictor effect lags in a functional linear model using basis expansions and penalized optimization.

## Key findings

- Effective estimation of predictor effect lags demonstrated
- Model shows strong predictive performance in simulations
- Mathematical properties of estimators are established

## Abstract

We propose a functional linear model to predict a response using multiple functional and longitudinal predictors and to estimate the effect lags of predictors. The coefficient functions are written as the expansion of a basis system (e.g. functional principal components, splines), and the coefficients of the fixed basis functions are estimated via optimizing a penalization criterion. Then time lags are determined by simultaneously searching on a prior grid mesh based on minimization of prediction error criterion. Moreover, mathematical properties of the estimated parameters and predicted responses are studied and performance of the method is evaluated by extensive simulations.