## Finite mixture regression: a sparse variable selection by model selection for clustering.(English)Zbl 1329.62279

Summary: We consider a finite mixture of Gaussian regression models for high-dimensional data, where the number of covariates may be much larger than the sample size. We propose to estimate the unknown conditional mixture density by a maximum likelihood estimator, restricted on relevant variables selected by an $$\ell_{1}$$-penalized maximum likelihood estimator. We get an oracle inequality satisfied by this estimator with a Jensen-Kullback-Leibler type loss. Our oracle inequality is deduced from a general model selection theorem for maximum likelihood estimators on a random model subcollection. We can derive the penalty shape of the criterion, which depends on the complexity of the random model collection.

### MSC:

 62H30 Classification and discrimination; cluster analysis (statistical aspects)
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### References:

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