Pointwise adaptive estimation of the marginal density of a weakly dependent process

- A data-driven procedure based on Goldenshluger-Lepski method is proposed.
- Considering pointwise risk allows us to select local bandwidths.
- Adaptive rates of convergence are obtained in several situations of dependence.
- Our procedure satisfies an oracle-type inequality.

This paper is devoted to the estimation of the common marginal density function of weakly dependent processes. The accuracy of estimation is measured using pointwise risks. We propose a data-driven procedure using kernel rules. The bandwidth is selected using the approach of Goldenshluger and Lepski and we prove that the resulting estimator satisfies an oracle type inequality. The procedure is also proved to be adaptive (in a minimax framework) over a scale of Hölder balls for several types of dependence: strong mixing processes, λ-dependent processes or i.i.d. sequences can be considered using a single procedure of estimation. Some simulations illustrate the performance of the proposed method.