We propose and study two pratically tractable methods to estimate the order of non-parametric hidden Markov models in addition to their parameters. The first one relies on estimating the rank of a matrix derived from the law of two consecutive observations while the second one selects the order by minimizing a penalized least squares criterion. We show strong consistency of both methods and prove an oracle inequality on the least squares estimators of the model parameters. We numerically compare their ability to select the right order in several situations and discuss their algorithmic complexity.
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