  Constrained approaches to maximum likelihood estimation in the context of
finite mixtures of normals have been presented in the literature. A fully
data-dependent constrained method for maximum likelihood estimation of
clusterwise linear regression is proposed, which extends previous work in
equivariant data-driven estimation of finite mixtures of Gaussians for
classification. The method imposes plausible bounds on the component variances,
based on a target value estimated from the data, which we take to be the
homoscedastic variance. Nevertheless, the present work does not only focus on
classification recovery, but also on how well model parameters are estimated.
In particular, the paper sheds light on the shrinkage-like interpretation of
the procedure, where the target is the homoscedastic model: this is not only
related to how close to the target the estimated scales are, but extends to the
estimated clusterwise linear regressions and classification. We show, based on
simulation and real-data based results, that our approach yields a final model
being the most appropriate-to-the-data compromise between the heteroscedastic
model and the homoscedastic model.
