Density estimation for $\beta$-dependent sequences

TL;DR

This paper investigates the performance of classical density estimators for stationary sequences, including non-mixing and long-range dependent data, providing new probabilistic tools and convergence rates for histogram estimators.

Contribution

It introduces a new Rosenthal-type inequality for BV functions and applies it to derive convergence rates of histograms for invariant densities of certain expanding maps.

Findings

New Rosenthal-type inequality for BV functions

Convergence rates for histogram estimators in complex dependent sequences

Applicability to non-mixing and long-range dependent data

Abstract

We study the Lp-integrated risk of some classical estimators of the density, when the observations are drawn from a strictly stationary sequence. The results apply to a large class of sequences, which can be non-mixing in the sense of Rosenblatt and long-range dependent. The main probabilistic tool is a new Rosenthal-type inequality for partial sums of BV functions of the variables. As an application, we give the rates of convergence of regular Histograms, when estimating the invariant density of a class of expanding maps of the unit interval with a neutral fixed point at zero. These Histograms are plotted in the section devoted to the simulations.