Unsupervised Learning Under a General Semiparametric Clusterwise Elliptical Distribution: Efficient Estimation, Optimal Clustering, and Consistent Cluster Selection

Abstract

A general semiparametric clusterwise elliptical distribution is introduced for assessing how latent cluster structure shapes continuous outcomes. Using a subjectwise representation, cluster-specific mean vectors are estimated by minimizing a weighted sum of squares criterion augmented with a separation penalty. An initialization scheme based on iterative Mahalanobis reassignment and a computational algorithm with guaranteed convergence accompany the estimation procedure. Under the localization conditions, the separation penalty criterion admits a local oracle property, and the corresponding localized partition recovers the realized latent cluster partition. The practical iterative Mahalanobis reassignment does not impose these truth-centered neighborhoods and is evaluated separately through numerical studies. The resulting first-phase partition seeds a second phase of pseudo-likelihood-based estimation and cluster reassignment. A semiparametric information criterion is also proposed for selecting the number of clusters. Monte Carlo simulations identify settings in which the proposed procedures improve on the evaluated parametric benchmarks, while additional experiments reveal sensitivity to cluster imbalance and departures from ellipticity. Applications illustrate the resulting customer and patient profiles.

arXiv comment56 pages, 4 figure