  Consider an i.i.d. sequence of random variables whose distribution f* lies in
one of a nested family of models M_q, q>=1. The smallest index q* such that
M_{q*} contains f* is called the model order. We establish strong consistency
of the penalized likelihood order estimator in a general setting with penalties
of order \eta(q) log log n, where \eta(q) is a dimensional quantity. Moreover,
such penalties are shown to be minimal. In contrast to previous work, an a
priori upper bound on the model order is not assumed. The results rely on a
sharp characterization of the pathwise fluctuations of the generalized
likelihood ratio statistic under entropy assumptions on the model classes. Our
results are applied to the geometrically complex problem of location mixture
order estimation, which is widely used but poorly understood.
