Pointwise Adaptive Estimation of the MarginalDensity of a Weakly Dependent Process

TL;DR

This paper introduces an adaptive, data-driven kernel-based method for estimating the marginal density of weakly dependent processes, achieving near-optimal accuracy across various dependence structures.

Contribution

It develops a unified, adaptive bandwidth selection procedure for kernel density estimation applicable to multiple dependence types, with proven theoretical guarantees.

Findings

Estimator satisfies an oracle inequality

Method adapts minimax rates over H"older classes

Simulations demonstrate effective performance

Abstract

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 datadriven 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 H\"older balls for several types of dependence: stong mixing processes, $\lambda$-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.