Colloquium Speaker:
Arash Foroushani, Ph.D. Candidate
University of Windsor
Title: "On matrix-valued estimators in Gaussian samples with high-dimensional data", joint work with Dr. Sévérien Nkurunziza
Abstract: Shrinkage estimation provides a powerful framework for constructing estimators of the normal mean that dominate the maximum likelihood estimator (MLE) under quadratic loss. Among the most prominent examples are shrinkage estimators (SEs), widely recognized for their risk-reduction properties. When observations arise from a non-singular distribution with known covariance structure, such estimators depend on the inverse of the covariance matrix. In low-dimensional settings with unknown covariance, this matrix is typically replaced by a suitable estimator. However, in high-dimensional settings, this approach encounters substantial theoretical and practical challenges. In particular, when the number of parameters exceeds the sample size, the sample covariance matrix becomes singular, rendering classical shrinkage procedures inapplicable. For vector-valued parameters, this issue has been studied by Chételat and Wells (2012, Annals of Statistics). Nevertheless, one of their central theorems, along with its proof, contains a flaw. We address this issue by providing a corrected version of the theorem together with a rigorous proof. Building on this foundation, we extend the methodology to matrix-valued parameters and investigate conditions ensuring the finiteness of the associated risk function. We also derive sufficient conditions under which the proposed SEs dominate the MLE. To complement the theoretical developments, we conduct simulation studies that demonstrate the effectiveness of the proposed estimators and support our theoretical findings. An additional key contribution of this work is the development of a unified shrinkage framework applicable in both low- and high-dimensional regimes.
Day & Time: Friday, April 24, 2026, at 3:00pm
Location: Lambton Tower, Room 9-118
Counts toward seminar attendance for MSc and PhD students in Math & Stats.