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Adaptive nonparametric estimation in the presence of dependence

publié le

Vendredi 3 avril : Nicolas Asin

(Université catholique de Louvain)

Adaptive nonparametric estimation in the presence of dependence

We consider nonparametric estimation problems in the presence of
dependent data, notably nonparametric regression with random design
and nonparametric density estimation. The proposed estimation
procedure is based on a dimension reduction. The minimax optimal
rate of convergence of the estimator is derived assuming a
sufficiently weak dependence characterized by fast decreasing mixing
coefficients. We illustrate these results by considering classical smoothness
assumptions. However, the proposed estimator requires an optimal
choice of a dimension parameter depending on certain characteristics
of the function of interest, which are not known in practice. The
main issue addressed in our work is an adaptive choice of this
dimension parameter combining model selection and Lepski’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 bound up to a constant provided
a fast decay of the mixing coefficients.