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

We introduce a general semiparametric clusterwise elliptical distribution to assess how latent cluster structure shapes continuous outcomes. Using a subjectwise representation, we first estimate cluster-specific mean vectors and a cluster-invariant scatter matrix by minimizing a weighted sum of squares criterion augmented with a separation penalty; we provide an initialization scheme and a computational algorithm with guaranteed convergence. This initial estimator consistently recovers the true clusters and seeds a second phase that alternates pseudo-maximum likelihood (or pseudo-maximum marginal likelihood) estimation with cluster reassignment, yielding asymptotic semiparametric efficiency and an optimal clustering that asymptotically maximizes the probability of correct membership. We also propose a semiparametric information criterion for selecting the number of clusters. Monte Carlo simulations and empirical applications demonstrate strong finite-sample performance and practical value.