Robust clustering in regression analysis via the contaminated Gaussian cluster-weighted model

The Gaussian cluster-weighted model (CWM) is a mixture of regression models with random covariates that allows for flexible clustering of a random vector composed of response variables and covariates. In each mixture component, a Gaussian distribution is adopted for both the covariates and the responses given the covariates. To make the approach robust with respect to the presence of mildly atypical observations, the contaminated Gaussian CWM is introduced. In addition to the parameters of the Gaussian CWM, each mixture component has a parameter controlling the proportion of outliers, one controlling the proportion of leverage points, one specifying the degree of contamination with respect to the response variables, and another specifying the degree of contamination with respect to the covariates. Crucially, these parameters do not have to be specified a priori, adding flexibility to the approach. Furthermore, once the model is estimated and the observations are assigned to the components, a finer intra-group classification in typical points, (mild) outliers, good leverage points, and bad leverage points—concepts of primary importance in robust regression analysis—can be directly obtained. Relations with other mixture-based contaminated models are analyzed, identifiability conditions are provided, an expectation-conditional maximization algorithm is outlined for parameter estimation, and various implementation and operational issues are discussed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo experiments and compared with other procedures. A sensitivity study is also conducted based on a real data set.