An adaptive method to robustify ML estimation in Cluster Weighted Modeling

Cluster-Weighted Models are a wide family of mixture distributions formodeling the joint probability of data coming from a heterogeneous population, and includes mixtures of distributions and mixtures of regressions as special cases.Unfortunately, they suffer from non-regularmaximum likelihood issues, due to possible spikes and unboundedness in the target function. We propose an improved version of the Gaussian Cluster-Weighted estimation methodology, by trimming aportion alpha of the data and imposing constraints to the estimated variances. Trimming provides robustness properties to the estimators and constraintsmove the maximizationproblemto a well-posed setting and allow to avoid spurious solutions, i.e. fitting a small localized random pattern in the data rather than a proper underlying cluster structure. Theoretical results are illustrated using a few empirical studies.