Component selection and variable selection for mixture regression models

Finite mixture regression models are commonly used to account for heterogeneity in populations and situations where the assumptions required for standard regression models may not hold. To expand the range of applicable distributions for components beyond the Gaussian distribution, other distributions, such as the exponential power distribution, the skew-normal distribution, and so on, are explored. To enable simultaneous model estimation, order selection, and variable selection, a penalized likelihood estimation approach that imposes penalties on both the mixing proportions and regression coefficients, which we call the double-penalized likelihood method is proposed in this paper. Four double-penalized likelihood functions and their performance are studied. The consistency of estimators, order selection, and variable selection are investigated. A modified expectation-maximization algorithm is proposed to implement the double-penalized likelihood method. Numerical simulations demonstrate the effectiveness of our proposed method and algorithm. Finally, the results of real data analysis are presented to illustrate the application of our approach. Overall, our study contributes to the development of mixture regression models and provides a useful tool for model and variable selection.