On the minimal penalty for Markov order estimation

Details On the minimal penalty for Markov order estimation On the minimal penalty for Markov order estimation

2009-08-25

arxiv.org/abs/0908.3666v1

Probab. Th. Rel. Fields 150 (2011), pp. 709-738

Mathematics

Probability

29 pages

Scientific paper

10.1007/s00440-010-0290-y

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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