  Cluster analysis requires many decisions: the clustering method and the
implied reference model, the number of clusters and, often, several
hyper-parameters and algorithms' tunings. In practice, one produces several
partitions, and a final one is chosen based on validation or selection
criteria. There exist an abundance of validation methods that, implicitly or
explicitly, assume a certain clustering notion. Moreover, they are often
restricted to operate on partitions obtained from a specific method. In this
paper, we focus on groups that can be well separated by quadratic or linear
boundaries. The reference cluster concept is defined through the quadratic
discriminant score function and parameters describing clusters' size, center
and scatter. We develop two cluster-quality criteria called quadratic scores.
We show that these criteria are consistent with groups generated from a general
class of elliptically-symmetric distributions. The quest for this type of
groups is common in applications. The connection with likelihood theory for
mixture models and model-based clustering is investigated. Based on bootstrap
resampling of the quadratic scores, we propose a selection rule that allows
choosing among many clustering solutions. The proposed method has the
distinctive advantage that it can compare partitions that cannot be compared
with other state-of-the-art methods. Extensive numerical experiments and the
analysis of real data show that, even if some competing methods turn out to be
superior in some setups, the proposed methodology achieves a better overall
performance.
