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We consider non-parametric estimation problems in the presence of dependentdata, notably non-parametric regression with random design and non-parametricdensity estimation. The proposed estimation procedure is based on a dimensionreduction. The minimax optimal rate of convergence of the estimator is derivedassuming a sufficiently weak dependence characterized by fast decreasing mixingcoefficients. We illustrate these results by considering classical smoothnessassumptions. However, the proposed estimator requires an optimal choice of adimension parameter depending on certain characteristics of the function ofinterest, which are not known in practice. The main issue addressed in our workis an adaptive choice of this dimension parameter combining model selection andLepski's method. It is inspired by the recent work of Goldenshluger and Lepski(2011). We show that this data-driven estimator can attain the lower risk boundup to a constant provided a fast decay of the mixing coefficients.

Adaptive non-parametric estimation in the presence of dependence