# Adaptive regression with Brownian path covariate

## TL;DR

This paper introduces an adaptive estimation method for regression functions with continuous outcomes and Wiener process covariates, utilizing Wiener-Itô decomposition and data-driven selection to achieve optimal convergence rates.

## Contribution

It develops a new adaptive regression estimator for functional covariates based on Wiener-Itô decomposition, with proven minimax convergence rates and a data-driven selection procedure.

## Key findings

- Achieves minimax convergence rates for the regression function estimation.
- Provides an oracle inequality leading to adaptive estimation.
- Demonstrates the effectiveness of the proposed method on Wiener process covariates.

## Abstract

This paper deals with estimation with functional covariates. More precisely, we aim at estimating the regression function $m$ of a continuous outcome $Y$ against a standard Wiener coprocess $W$. Following Cadre and Truquet (2015) and Cadre, Klutchnikoff, and Massiot (2017) the Wiener-It\^o decomposition of $m(W)$ is used to construct a family of estimators. The minimax rate of convergence over specific smoothness classes is obtained. A data-driven selection procedure is defined following the ideas developed by Goldenshluger and Lepski (2011). An oracle-type inequality is obtained which leads to adaptive results.