The Annals of Statistics
- Ann. Statist.
- Volume 39, Number 4 (2011), 1852-1877.
Optimal model selection for density estimation of stationary data under various mixing conditions
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.
Ann. Statist., Volume 39, Number 4 (2011), 1852-1877.
First available in Project Euclid: 26 July 2011
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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
- 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).