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On the minimal penalty for Markov order estimation

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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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  1. 1

    Cappé, O., Moulines, E., Rydén, T.: Inference in hidden Markov models. Springer Series in Statistics. Springer, New York (2005) (With Randal Douc’s contributions to Chapter 9 and Christian P. Robert’s to Chapters 6, 7 and 13, With Chapter 14 by Gersende Fort, Philippe Soulier and Moulines, and Chapter 15 by Stéphane Boucheron and Elisabeth Gassiat)

  2. 2

    Csiszár, I.: Large-scale typicality of Markov sample paths and consistency of MDL order estimators. IEEE Trans. Inform. Theory, 48(6), 1616–1628 (2002) (Special issue on Shannon theory: perspective, trends, and applications)

  3. 3

    Csiszár I., Shields P.C.: The consistency of the BIC Markov order estimator. Ann. Stat. 28(6), 1601–1619 (2000)

  4. 4

    Csiszár I., Talata Z.: Context tree estimation for not necessarily finite memory processes, via BIC and MDL. IEEE Trans. Inform. Theory 52(3), 1007–1016 (2006)

  5. 5

    Finesso, L.: Consistent estimation of the order for Markov and hidden Markov chains. Ph.D. thesis, University of Maryland (1990)

  6. 6

    Kieffer J.C.: Strongly consistent code-based identification and order estimation for constrained finite-state model classes. IEEE Trans. Inform. Theory 39(3), 893–902 (1993)

  7. 7

    Massart, P.: Concentration inequalities and model selection. Lecture Notes in Mathematics, vol. 1896. Springer, Berlin (2007) (Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003, With a foreword by Jean Picard)

  8. 8

    van de Geer S.A.: Applications of Empirical Process Theory. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 6. Cambridge University Press, Cambridge (2000)

  9. 9

    van der Vaart, A.W., Wellner, J.A.: Weak convergence and empirical processes. Springer Series in Statistics. Springer, New York (1996) (With applications to statistics)

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Correspondence to Ramon van Handel.

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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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  • 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