Consistent order estimation and minimal penalties

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