Adaptive Regression With Brownian Path Covariate

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 (ESAIM Probab. Stat. 19 (2015) 251-267) and Cadre et al. (ESAIM Probab. Stat. 21 (2017) 138-158) the Wiener-Ito 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 (Ann. Statist. 39 (2011) 1608-1632). An oracle-type inequality is obtained which leads to adaptive results.