Abstract
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.
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