## Electronic Journal of Statistics

### Density estimation for $\tilde{\beta}$-dependent sequences

#### Abstract

We study the ${\mathbb{L}}^{p}$-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.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 981-1021.

Dates
First available in Project Euclid: 30 March 2017

https://projecteuclid.org/euclid.ejs/1490860814

Digital Object Identifier
doi:10.1214/17-EJS1249

Mathematical Reviews number (MathSciNet)
MR3629417

Zentralblatt MATH identifier
1362.62079

Subjects
Primary: 62G07: Density estimation
Secondary: 60G10: Stationary processes

#### Citation

Dedecker, Jérôme; Merlevède, Florence. Density estimation for $\tilde{\beta}$-dependent sequences. Electron. J. Statist. 11 (2017), no. 1, 981--1021. doi:10.1214/17-EJS1249. https://projecteuclid.org/euclid.ejs/1490860814

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