# Maxiset for model selection

### joint work with F.  Autin, J.-M. Loubes and V. Rivoirard

What are the functions well estimated by model selection techniques? In this work, we consider the Gaussian white noise model and estimators by projection on spaces spanned by elements of a dictionary. Let $m$ be such a space, the estimator $\widehat{s}_m$ associated to this model for a noise variance of $\frac{1}{n}$ by $$\widehat{s}_{m} = \mathop{\mathrm{argmin}}_{s \in \mathcal{M}_n}\ \gamma_n(s)$$ where $\gamma_n$ is the usual contrast. Let $\mathcal{M}_n$ be a model collection and $\mathop{\mathrm{pen}}_n(m)$ a penalty family for these models, the penalized estimator $\widehat{s}_{\widehat{m}}$ is defined as the previous projection estimator in the model $\widehat{m}$ minimizing $$\gamma_n(\widehat{s}_m) + \mathop{\mathrm{pen}}_n(m).$$

We assume that the penalty is such that for all $n$ $$\mathbb{E}[ \|s_0-\widehat{s}_{\widehat{m}}\|^2 ] \leq C \inf_{m \in \mathcal{M}_n}\ \inf_{s \in \mathcal{M}_n} \|s_0-s\|^2 + \mathop{\mathrm{pen}}_n(m) + \frac{c}{n}.$$ Under weak assumptions on the penalties, the structure of the model collections $\mathcal{M}_n$ and the rates $\rho_n$, we prove the equivalence between $$(\rho_n)^{-2} \mathbb{E}[ \|s_0-\widehat{s}_{\widehat{m}}\|^2 ] < +\infty \quad\text{ and }\quad (\rho_n)^{-2} \left( \inf_{m \in \mathcal{M}_n}\ \inf_{s \in \mathcal{M}_n} \|s_0-s\|^2 + \mathop{\mathrm{pen}}_n(m) \right)< +\infty.$$ The functions that are well estimated are exactly those that are well approximated!

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