  When a population exhibits heterogeneity, we often model it via a finite
mixture: decompose it into several different but homogeneous subpopulations.
Contemporary practice favors learning the mixtures by maximizing the likelihood
for statistical efficiency and the convenient EM-algorithm for numerical
computation. Yet the maximum likelihood estimate (MLE) is not well defined for
the most widely used finite normal mixture in particular and for finite
location-scale mixture in general. We hence investigate feasible alternatives
to MLE such as minimum distance estimators. Recently, the Wasserstein distance
has drawn increased attention in the machine learning community. It has
intuitive geometric interpretation and is successfully employed in many new
applications. Do we gain anything by learning finite location-scale mixtures
via a minimum Wasserstein distance estimator (MWDE)? This paper investigates
this possibility in several respects. We find that the MWDE is consistent and
derive a numerical solution under finite location-scale mixtures. We study its
robustness against outliers and mild model mis-specifications. Our moderate
scaled simulation study shows the MWDE suffers some efficiency loss against a
penalized version of MLE in general without noticeable gain in robustness. We
reaffirm the general superiority of the likelihood based learning strategies
even for the non-regular finite location-scale mixtures.
