Hidden Markov Models with mixtures as emission distributions

In unsupervised classification, Hidden Markov Models (HMM) are used to account for a neighborhood structure between observations. The emission distributions are often supposed to belong to some parametric family. In this paper, a semiparametric modeling where the emission distributions are a mixture of parametric distributions is proposed to get a higher flexibility. We show that the classical EM algorithm can be adapted to infer the model parameters. For the initialisation step, starting from a large number of components, a hierarchical method to combine them into the hidden states is proposed. Three likelihood-based criteria to select the components to be combined are discussed. To estimate the number of hidden states, BIC-like criteria are derived. A simulation study is carried out both to determine the best combination between the merging criteria and the model selection criteria and to evaluate the accuracy of classification. The proposed method is also illustrated using a biological dataset from the model plant Arabidopsis thaliana. A R package HMMmix is freely available on the CRAN.

A leave-p-out based estimation of the proportion of null hypotheses

In the multiple testing context, a challenging problem is the estimation of the proportion $pi_0$ of true-null hypotheses. A large number of estimators of this quantity rely on identifiability assumptions that either appear to be violated on real data, or may be at least relaxed. Under independence, we propose an estimator $hat{pi}_0$ based on density estimation using both histograms and cross-validation. Due to the strong connection between the false discovery rate (FDR) and $pi_0$, many multiple testing procedures (MTP) designed to control the FDR may be improved by introducing an estimator of $pi_0$. We provide an example of such an improvement (plug-in MTP) based on the procedure of Benjamini and Hochberg. Asymptotic optimality results may be derived for both $hat{pi}_0$ and the resulting plug-in procedure. The latter ensures the desired asymptotic control of the FDR, while it is more powerful than the BH-procedure. Finally, we compare our estimator of $pi_0$ with other widespread estimators in a wide range of simulations. We obtain better results than other tested methods in terms of mean square error (MSE) of the proposed estimator. Finally, both asymptotic optimality results and the interest in tightly estimating $pi_0$ are confirmed (empirically) by results obtained with the plug-in MTP.