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–Itô 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.