Pathwise fluctuations of likelihood ratios and consistent order estimation

Elisabeth Gassiat (1) , Ramon van Handel (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Operations Research & Financial Engineering

Consider an i.i.d. sequence of random variables whose distribution lies in one of a nested family of models M_q. We obtain a sharp characterization of the pathwise fluctuations of the generalized likelihood ratio statistic under entropy assumptions on the model classes M_q. Moreover, we develop a technique to obtain local entropy bounds from global entropy computations, so that these results can be applied in models with non-regular geometric structure. Finally, the results are applied to prove strong consistency and to identify minimal penalties for penalized likelihood order estimators in the absence of prior upper bounds on the model order and the underlying parameter set. Location mixture models, which possess a notoriously complicated geometric structure, are used as a case study throughout the paper, and the requisite geometric analysis is of independent interest.

Mots clés: uniform law of iterated logarithm; local entropy with bracketing; entropy of finite mixtures; penalized likelihood order estimation; strong consistency

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Elisabeth Gassiat, Ramon van Handel. Pathwise fluctuations of likelihood ratios and consistent order estimation. 2011. ⟨hal-00453469v2⟩