All Times ET
Keywords: Stein’s paradox, James-Stein estimation, Sample eigenvectors, PCA
Stein’s paradox, which loosely states that there is a better way to estimate several quantities simultaneously than to estimate each individually, is one of the most surprising facts in all of statistics. The resulting James-Stein estimators have been reinvented in many different forms and in numerous applications. In recent work, Goldberg, Papanicalaou & Shkolnik (GPS, 2021) supply geometrically motivated corrections of finite-sample bias residing in the principal components of a covariance matrix. This was previously not thought possible, as the characterization of bias in a high-dimensional space is challenging. The solution in GPS (2021) amounts to choosing the right coordinate system and applying the spherical law of cosines. In this work, we show that the correction of GPS (2021) is closely tied to James- Stein estimation. The connections are revealed after changing coordinates from a unit sphere to a hyperplane with a dispersionless normal vector. We formalize these connections by relating the problem of estimating eigenvectors for randomly perturbed matrices to the classic mean-estimation problem initially considered by Stein.