Estimating beta-mixing coefficients via histograms

TL;DR

This paper introduces the first estimator for beta-mixing coefficients from a single stationary time series sample, using histogram density estimates and Markov approximations, with proven consistency and convergence rates.

Contribution

It provides a novel, risk-consistent estimator for beta-mixing coefficients based on finite data and Markov approximation, filling a gap in statistical learning for time series.

Findings

Estimator is risk consistent and converges as sample size increases.

Convergence rates are established for the Markov approximation.

Method demonstrated through simulations and real data.

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.