Asymptotics of Markov Order Estimators for Infinite Memory Processes

• Activity Number: 352
• Type: Contributed
• Date/Time: Tuesday, August 2, 2011 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Zsolt Talata*+
• Companies: University of Kansas
• Keywords: Markov order estimator ; information criterion ; divergence rate ; asymptotics ; ergodic process ; infinite memory

Abstract

For finite-alphabet stationary ergodic processes with infinite memory, Markov order estimators that optimize an information criterion over the candidate orders based on a sample of size n are investigated. Three familiar information criteria are considered: the Bayesian information criterion (BIC) with generalized penalty term yielding the penalized maximum likelihood (PML), and the normalized maximum likelihood (NML) and the Krichevsky-Trofimov (KT) code lengths. A bound on the probability that the estimated order is greater than some order is obtained under the assumption that the process is weakly non-null and alpha-summable. This gives an O(log n) upper bound on the estimated order eventually almost surely as n tends to infinity. Moreover, a bound on the probability that the estimated order is less than some order is obtained if the decay of the continuity rate of the weakly non-null process is in some exponential range. This implies that then the estimated order attains the O(log n) divergence rate eventually almost surely as n tends to infinity.