## $$\ell_{1}$$-penalization for mixture regression models.(English)Zbl 1203.62128

Summary: We consider a finite mixture of regressions (FMR) model for high-dimensional inhomogeneous data where the number of covariates may be much larger than sample size. We propose an $$\ell _{1}$$-penalized maximum likelihood estimator in an appropriate parameterization. This kind of estimation belongs to a class of problems where optimization and theory for non-convex functions is needed. This distinguishes itself very clearly from high-dimensional estimation with convex loss- or objective functions as, for example, with the Lasso in linear or generalized linear models. Mixture models represent a prime and important example where non-convexity arises. For FMR models, we develop an efficient EM algorithm for numerical optimization with provable convergence properties. Our penalized estimator is numerically better posed (e.g., boundedness of the criterion function) than unpenalized maximum likelihood estimation, and it allows for effective statistical regularization including variable selection. We also present some asymptotic theory and oracle inequalities: due to non-convexity of the negative log-likelihood function, different mathematical arguments are needed than for problems with convex losses. Finally, we apply the new method to both simulated and real data.

### MSC:

 62J12 Generalized linear models (logistic models) 62F12 Asymptotic properties of parametric estimators 62J07 Ridge regression; shrinkage estimators (Lasso) 90C90 Applications of mathematical programming 65C60 Computational problems in statistics (MSC2010)

flexmix; glmnet
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