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June 2006 Testing the order of a model
Antoine Chambaz
Ann. Statist. 34(3): 1166-1203 (June 2006). DOI: 10.1214/009053606000000344


This paper deals with order identification for nested models in the i.i.d. framework. We study the asymptotic efficiency of two generalized likelihood ratio tests of the order. They are based on two estimators which are proved to be strongly consistent. A version of Stein’s lemma yields an optimal underestimation error exponent. The lemma also implies that the overestimation error exponent is necessarily trivial. Our tests admit nontrivial underestimation error exponents. The optimal underestimation error exponent is achieved in some situations. The overestimation error can decay exponentially with respect to a positive power of the number of observations.

These results are proved under mild assumptions by relating the underestimation (resp. overestimation) error to large (resp. moderate) deviations of the log-likelihood process. In particular, it is not necessary that the classical Cramér condition be satisfied; namely, the log-densities are not required to admit every exponential moment. Three benchmark examples with specific difficulties (location mixture of normal distributions, abrupt changes and various regressions) are detailed so as to illustrate the generality of our results.


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Antoine Chambaz. "Testing the order of a model." Ann. Statist. 34 (3) 1166 - 1203, June 2006.


Published: June 2006
First available in Project Euclid: 10 July 2006

zbMATH: 1096.62016
MathSciNet: MR2278355
Digital Object Identifier: 10.1214/009053606000000344

Primary: 60F10 , 60G57 , 62C99 , 62F03 , 62F05 , 62F12

Keywords: Abrupt changes , Empirical processes , error exponents , Hypothesis testing , large deviations , mixtures , Model selection , Moderate deviations , order estimation

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 3 • June 2006
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