# Kwun Chuen Gary Chan (University of Washington)

Date

Friday February 3, 20232:30 pm - 3:30 pm

Location

Jeffery Hall, Room 234## Math & Stats Department Colloquium

**Friday, February 3rd, 2023**

**Time:** 2:30 p.m. **Place:** Jeffery Hall, Room 234

**Speaker:** Kwun Chuen Gary Chan (University of Washington)

**Title:** Stein shrinkage, random matrix and imaginary direction smoothing

**Abstract:** The Jame-Stein estimator of a multivariate mean vector and the Stein estimator of a covariance matrix are classical examples of shrinkage estimators. Conceptually very different from the majority of contemporary estimators formulated using penalization with sparsity and/or low-rank assumptions, Stein covariance estimator targets non-sparse and full-rank covariance matrices. We review recent literature in random matrix theory which are relevant in studying Stein-type shrinkage estimators and extensions, and present certain asymptotic optimality results under various loss functions. A recurring unknown quantity to be estimated is a real-direction limit of a Stieltjes transform of the limiting spectral distribution, defined on the upper complex half-plane. Moment-based estimators of such quantities are quite unstable but were used in Stein's original paper. We consider an alternative estimator by defining a smoothed loss function based on smoothed empirical spectral distribution, with an optimum in terms of a smoothed Stieljes transform, where smoothing is along the imaginary direction via a Cauchy kernel. A data-adaptive choice of the smoothing parameter is proposed due to a relationship between the imaginary part of a complex Stieltjes transform and the sample eigenvalue distribution. Some theoretical properties and numerical illustration will be discussed. This is a joint work with Sheung Chi Phillip Yam, Xiaolong Li and Yifan Shi.

**Gary Chan** is a professor jointly appointed in the Department of Biostatistics and Department of Health Systems and Population Health at the University of Washington. He has a broad statistical research interest on observational data, including complex designs, exposures and outcomes. He also has theoretical interest in certain semiparametric and nonparametric problems. He is a Fellow of the American Statistical Association and Institute of Mathematical Statistics. He is currently an associate editor of Journal of American Statistical Association (Theory and Methods) as well as a few statistics and public health journals.