Nonparametric contaminated Gaussian mixture of regressions

Semi and nonparametric mixture of regressions are a very useful flexible class of mixture of regressions in which some or all of the parameters are nonparametric functions of the covariates. These models typically assume that the conditional distribution of each component is Gaussian. However, observed data are usually contaminated by outliers which may lead to poor performance of the estimation procedure. In this paper, we propose semi and nonparametric contaminated Gaussian mixture of regressions to robustly estimate the parametric and/or nonparametric terms of the models in the presence of mild outliers. The latter are data points that do not deviate from the regression relationship within a given component but they produce an overall component distribution that is too heavy-tailed to be modelled by a Gaussian distribution. The virtue of using a contaminated Gaussian error distribution is that we can simultaneously perform model-based clustering of observations and model-based outlier detection. We propose two algorithms, an expectation-maximization (EM)-type algorithm and an expectation-conditional-maximization (ECM)-type algorithm, to perform maximum likelihood and local-likelihood kernel estimation of the parametric and nonparametric terms of the proposed models, respectively. The robustness of the proposed models is examined using an extensive simulation study. The practical utility of the proposed models is demonstrated using two real economic datasets. The code for fitting the proposed models in this paper can be found through the link: Fit_NPCGMRs_model .