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On the minimal penalty for Markov order estimation
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  • Published: 14 April 2010

On the minimal penalty for Markov order estimation

  • Ramon van Handel1 

Probability Theory and Related Fields volume 150, pages 709–738 (2011)Cite this article

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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.

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Author information

Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, 08544, USA

    Ramon van Handel

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  1. Ramon van Handel
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Correspondence to Ramon van Handel.

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Cite this article

van Handel, R. On the minimal penalty for Markov order estimation. Probab. Theory Relat. Fields 150, 709–738 (2011). https://doi.org/10.1007/s00440-010-0290-y

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  • Received: 26 August 2009

  • Revised: 22 February 2010

  • Published: 14 April 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0290-y

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Keywords

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

Mathematics Subject Classification (2000)

  • Primary 62M05
  • Secondary 60E15
  • 60F15
  • 60G42
  • 60J10
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