## Electronic Journal of Statistics

### Estimating beta-mixing coefficients via histograms

#### 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.

#### Article information

Source
Electron. J. Statist. Volume 9, Number 2 (2015), 2855-2883.

Dates
First available in Project Euclid: 31 December 2015

https://projecteuclid.org/euclid.ejs/1451577417

Digital Object Identifier
doi:10.1214/15-EJS1094

Mathematical Reviews number (MathSciNet)
MR3439187

Zentralblatt MATH identifier
1330.62344

#### Citation

McDonald, Daniel J.; Shalizi, Cosma Rohilla; Schervish, Mark. Estimating beta-mixing coefficients via histograms. Electron. J. Statist. 9 (2015), no. 2, 2855--2883. doi:10.1214/15-EJS1094. https://projecteuclid.org/euclid.ejs/1451577417.

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