Adaptive Estimation in Autoregression or Β-mixing Regression via Model Selection

We study the problem of estimating some unknown regression function in a β-mixing dependent framework. To this end, we consider some collection of models which are finite dimensional spaces. A penalized leastsquares estimator (PLSE) is built on a data driven selected model among this collection. We state non asymptotic risk bounds for this PLSE and give several examples where the procedure can be applied (autoregression, regression with arithmetically β-mixing design points, regression with mixing errors, estimation in additive frameworks, estimation of the order of the autoregression ...). In addition we show that under a weak moment condition on the errors, our estimator is adaptive in the minimax sense simultaneously over some family of Besov balls.