Testing the order of a model

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 Cramer 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.