A New Approach to Estimate Semi-Parametric Gaussian Mixtures of Regressions with Varying Mixing Proportions

The semi-parametric Gaussian mixture of regressions with varying proportions (SPGMRVPs) model is a flexible version of a Gaussian mixture of linear regressions (GMLRs) model. The model assumes that the mixing probabilities are non-parametric functions of the covariate(s) whereas the component regression functions are parametric and the variances are constants. However, this flexibility is not without its challenges. Traditional methods of estimation are not guaranteed to produce reliable estimates of the model. A local-likelihood approach for estimating the non-parametric functions requires that we maximize a set of local-likelihood functions. Using the Expectation-Maximization (EM) algorithm to separately maximize each local-likelihood function may lead to label switching. This is because the responsibilities calculated at each local E-step are not guaranteed to be aligned. The consequence of this label-switching is wiggly and non-smooth non-parametric functions. In this chapter, we propose a unified approach to address label-switching and obtain sensible estimates. We propose a model-based approach to address the label-switching problem. Locally, the SPGMRVPs model is a GMLRs. Thus, we reformulate the SPGMRVPs model as a mixture of these GMLRs. Estimating the mixture of GMLRs is equivalent to simultaneously maximizing the local-likelihood functions. The estimation is carried out using the classical EM algorithm. Next, we propose one-step backfitting estimates of the parametric and non-parametric terms. The effectiveness and practical utility of the proposed approach is demonstrated using Monte Carlo simulations and an environmental data analysis, respectively.