  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
Cram\'{e}r 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.
