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Is model selection possible for the $\ell_p$-loss? PCO estimation for regression models
Claire Lacour, Pascal Massart, Vincent Rivoirard · 2025-04-15 · via math.ST updates on arXiv.org

This paper addresses the problem of model selection in the sequence model $Y=θ+\varepsilonξ$, when $ξ$ 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, $p\geq 1.$ 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.

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