2016/02/01 by Nicolas Asin, Asin, Nicolas, Jan Johannes +1 · 1 citation
Computer Science · Mathematics · #62G08 #Bayesian Methods and Mixture Models #FOS: Mathematics #Primary 62G05 #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST) #secondary 62G07
paper · pdf · doi:10.48550/arxiv.1602.00531
openalex publication_date 2016/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.