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2015 Estimating beta-mixing coefficients via histograms
Daniel J. McDonald, Cosma Rohilla Shalizi, Mark Schervish
Electron. J. Statist. 9(2): 2855-2883 (2015). DOI: 10.1214/15-EJS1094

Abstract

The literature on statistical learning for time series often assumes asymptotic independence or “mixing” of the data-generating process. These mixing assumptions are never tested, nor are there methods for estimating mixing coefficients from data. Additionally, for many common classes of processes (Markov processes, ARMA processes, etc.) general functional forms for various mixing rates are known, but not specific coefficients. We present the first estimator for beta-mixing coefficients based on a single stationary sample path and show that it is risk consistent. Since mixing rates depend on infinite-dimensional dependence, we use a Markov approximation based on only a finite memory length $d$. We present convergence rates for the Markov approximation and show that as $d\rightarrow\infty$, the Markov approximation converges to the true mixing coefficient. Our estimator is constructed using $d$-dimensional histogram density estimates. Allowing asymptotics in the bandwidth as well as the dimension, we prove $L^{1}$ concentration for the histogram as an intermediate step. Simulations wherein the mixing rates are calculable and a real-data example demonstrate our methodology.

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Daniel J. McDonald. Cosma Rohilla Shalizi. Mark Schervish. "Estimating beta-mixing coefficients via histograms." Electron. J. Statist. 9 (2) 2855 - 2883, 2015. https://doi.org/10.1214/15-EJS1094

Information

Received: 1 December 2014; Published: 2015
First available in Project Euclid: 31 December 2015

zbMATH: 1330.62344
MathSciNet: MR3439187
Digital Object Identifier: 10.1214/15-EJS1094

Rights: Copyright © 2015 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.9 • No. 2 • 2015
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