Adaptive non-parametric estimation in the presence of dependence

TL;DR

This paper develops an adaptive non-parametric estimation method for dependent data, achieving near-optimal convergence rates by combining model selection and Lepski's method under weak dependence conditions.

Contribution

It introduces a novel adaptive procedure for non-parametric estimation with dependent data, addressing the challenge of unknown function characteristics.

Findings

Estimator attains minimax optimal rates under fast mixing conditions.

Adaptive method effectively selects the dimension parameter in practice.

Results extend classical smoothness assumptions to dependent data scenarios.

Abstract

We consider non-parametric estimation problems in the presence of dependent data, notably non-parametric regression with random design and non-parametric 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.