Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) with finite-valued latent process and statedependent distributions from a general one-parameter family. A test for m = 2 against m ≥ 3 states of the underlying Markov chain is proposed. So far, no satisfactory methods for this problem are available. The proposed test is an extension to HMMs of the modified likelihood ratio test (LRT) for two-states in a finite mixture, as introduced by Chen, Chen & Kalbfleisch (J. R. Stat. Soc. Ser. B 66, 2004, 95–115). The authors develop its asymptotic distribution theory under the null hypothesis of two states, and investigate its finite sample properties in a simulation study. The test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, it is also shown how to test general regular hypotheses on the marginal mixture of HMMs via a quasi LRT. Two empirical illustrations conclude the paper. Title in French: we can supply this Résumé : The authors consider hidden Markov models (HMMs) with finite-valued latent process and state-dependent distributions from a general one-parameter family. A test for m = 2 against m ≥ 3 states of the underlying Markov chain is proposed. So far, no satisfactory methods for this problem are available. The proposed test is an extension to HMMs of the modified likelihood ratio test (LRT) for two-states in a finite mixture, as introduced by Chen, Chen & Kalbfleisch (J. R. Stat. Soc. Ser. B 66, 2004, 95–115). The authors develop its asymptotic distribution theory under the null hypothesis of two states, and investigate its finite sample properties in a simulation study. The test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, it is also shown how to test general regular hypotheses on the marginal mixture of HMMs via a quasi LRT. Two empirical illustrations conclude the paper.