The Annals of Statistics

Optimal model selection for density estimation of stationary data under various mixing conditions

Matthieu Lerasle

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Abstract

We propose a block-resampling penalization method for marginal density estimation with nonnecessary independent observations. When the data are β or τ-mixing, the selected estimator satisfies oracle inequalities with leading constant asymptotically equal to 1.

We also prove in this setting the slope heuristic, which is a data-driven method to optimize the leading constant in the penalty.

Article information

Source
Ann. Statist., Volume 39, Number 4 (2011), 1852-1877.

Dates
First available in Project Euclid: 26 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1311688538

Digital Object Identifier
doi:10.1214/11-AOS888

Mathematical Reviews number (MathSciNet)
MR2893855

Zentralblatt MATH identifier
1227.62018

Subjects
Primary: 62G07: Density estimation 62G09: Resampling methods
Secondary: 62M99: None of the above, but in this section

Keywords
Density estimation optimal model selection resampling methods slope heuristic weak dependence

Citation

Lerasle, Matthieu. Optimal model selection for density estimation of stationary data under various mixing conditions. Ann. Statist. 39 (2011), no. 4, 1852--1877. doi:10.1214/11-AOS888. https://projecteuclid.org/euclid.aos/1311688538


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Supplemental materials

  • Supplementary material: Proofs of Lemmas 5.1 and 5.2. In the Supplementary Material, we give complete proofs of the concentrations Lemmas 5.1 and 5.2. We use coupling results, respectively, of Berbee (1979) and Dedecker and Prieur (2005), to build sequences of independent random variables (A_0^∗, …, A_(p−1)^∗) approximating the sequence of blocks (A_0, …, A_(p−1)), respectively in the β and τ mixing case. We prove concentration lemmas equivalent to Lemmas 5.1 and 5.2 for these approximating random variables. The main tools here are the concentration inequalities of Bousquet (2002) and Klein and Rio (2005) for the maximum of the empirical process. We prove finally some covariance inequalities to evaluate the expectation of p(m) and deduce the rates ε_n = (ln n)^(−1/2).