On the minimal penalty for Markov order estimation

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Abstract

We show that large-scale typicality of Markov sample paths implies that the likelihood ratio statistic satisfies a law of iterated logarithm uniformly to the same scale. As a consequence, the penalized likelihood Markov order estimator is strongly consistent for penalties growing as slowly as log log n when an upper bound is imposed on the order which may grow as rapidly as log n. Our method of proof, using techniques from empirical process theory, does not rely on the explicit expression for the maximum likelihood estimator in the Markov case and could therefore be applicable in other settings.

Original languageEnglish (US)
Pages (from-to)709-738
Number of pages30
JournalProbability Theory and Related Fields
Volume150
Issue number3
DOIs
StatePublished - Aug 1 2011

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Empirical process theory
  • Large-scale typicality
  • Markov chains
  • Martingale inequalities
  • Order estimation
  • Uniform law of iterated logarithm

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