Fractals with point impact in functional linear regression

TL;DR

This paper introduces a point impact linear regression model for continuous stochastic processes evaluated at a specific time, emphasizing interpretability and incorporating fractal properties like self-similarity.

Contribution

It proposes a new point impact model with fractal trajectories, develops bootstrap confidence intervals, and analyzes model misspecification effects.

Findings

Bootstrap confidence intervals for the sensitive time point are effective.

The model accounts for fractal properties with unknown Hurst exponent.

Misspecification impacts the distribution and convergence rates.

Abstract

This paper develops a point impact linear regression model in which the trajectory of a continuous stochastic process, when evaluated at a sensitive time point, is associated with a scalar response. The proposed model complements and is more interpretable than the functional linear regression approach that has become popular in recent years. The trajectories are assumed to have fractal (self-similar) properties in common with a fractional Brownian motion with an unknown Hurst exponent. Bootstrap confidence intervals based on the least-squares estimator of the sensitive time point are developed. Misspecification of the point impact model by a functional linear model is also investigated. Non-Gaussian limit distributions and rates of convergence determined by the Hurst exponent play an important role.