Adaptive estimation in circular functional linear models

TL;DR

This paper develops a data-driven adaptive estimator for the slope function in circular functional linear regression, achieving optimal minimax convergence rates without prior knowledge of the problem's ill-posedness.

Contribution

It introduces a fully adaptive orthogonal series estimator for the slope function in circular functional linear models, extending previous methods by removing the need for known ill-posedness degree.

Findings

Estimator attains the optimal minimax rate of convergence.

The method adapts to unknown degrees of ill-posedness.

It provides consistent estimation of the slope function and its derivatives.

Abstract

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of 1-periodic, second order stationary random functions X1,...,Xn. We consider an orthogonal series estimator of the slope function, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. Wepropose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill posedness to be known. Then we generalize the procedure to a random set of admissible m's and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in term of a general weighted L2-risk. This means that we provide adaptive estimators of both the slope function and its derivatives.