Is model selection possible for the ${\ell }_p$-loss? PCO estimation for regression models

Abstract

This paper addresses the problem of model selection in the sequence model $Y=\hat{I}¸+\epsilon \hat{I}¾$, when $\hat{I}¾$ is sub-Gaussian, for non-euclidian loss-functions. In this model, the Penalized Comparison to Overfitting procedure is studied for the weighted ${\ell }_p$-loss, $p\ge 1.$ Several oracle inequalities are derived from concentration inequalities for sub-Weibull variables. Using judicious collections of models and penalty terms, minimax rates of convergence are stated for Besov bodies ${\mathcal{B}}_{r,\infty }^s$. These results are applied to the functional model of nonparametric regression.