Adaptive density estimation for stationary processes

TL;DR

This paper introduces an adaptive density estimation algorithm for stationary processes with mixing conditions, achieving optimal convergence rates similar to independent data scenarios.

Contribution

It develops a model selection procedure based on a generalized Mallows' C_p for stationary processes with mixing, providing oracle inequalities and adaptivity over Besov spaces.

Findings

Estimator achieves i.i.d. convergence rates

Oracle inequalities established for the estimator

Method applicable to β- and τ-mixing processes

Abstract

We propose an algorithm to estimate the common density $s$ of a stationary process $X_1,...,X_n$. We suppose that the process is either $\beta$ or $\tau$-mixing. We provide a model selection procedure based on a generalization of Mallows' $C_p$ and we prove oracle inequalities for the selected estimator under a few prior assumptions on the collection of models and on the mixing coefficients. We prove that our estimator is adaptive over a class of Besov spaces, namely, we prove that it achieves the same rates of convergence as in the i.i.d framework.