Divergence rates of Markov order estimators and their application to statistical estimation of stationary ergodic processes

TL;DR

This paper analyzes the convergence rates of Markov order estimators for stationary ergodic processes and introduces generalized consistency concepts for infinite memory processes, with explicit bounds and applications.

Contribution

It provides explicit convergence rates for PML Markov order estimators and extends the notion of consistent order estimation to infinite memory processes.

Findings

Derived explicit convergence rates with constants.

Established bounds on estimated Markov order.

Generalized consistency for infinite memory processes.

Abstract

Stationary ergodic processes with finite alphabets are estimated by finite memory processes from a sample, an n-length realization of the process, where the memory depth of the estimator process is also estimated from the sample using penalized maximum likelihood (PML). Under some assumptions on the continuity rate and the assumption of non-nullness, a rate of convergence in $\bar{d}$-distance is obtained, with explicit constants. The result requires an analysis of the divergence of PML Markov order estimators for not necessarily finite memory processes. This divergence problem is investigated in more generality for three information criteria: the Bayesian information criterion with generalized penalty term yielding the PML, and the normalized maximum likelihood and the Krichevsky-Trofimov code lengths. Lower and upper bounds on the estimated order are obtained. The notion of consistent Markov order estimation is generalized for infinite memory processes using the concept of oracle order estimates, and generalized consistency of the PML Markov order estimator is presented.