Improved Shrinkage Prediction Under a Spiked Covariance Structure

• Activity Number: 170 - Theory and Methods for High-Dimensional Data
• Type: Contributed
• Date/Time: Monday, July 30, 2018 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Trambak Banerjee* and Gourab Mukherjee and Debashis Paul
• Companies: and University of Southern California and UC Davis
• Keywords: Shrinkage estimator; Asymptotic optimality; Spiked Covariance; Non-exchangeable priors; Quantile loss; Linex

Abstract

We develop a novel shrinkage rule for prediction in a high dimensional non-exchangeable hierarchical Gaussian model with an unknown spiked covariance structure. We propose a family of commutative priors which, governed by a power hyper-parameter, ranges from perfect independence to highly dependent scenarios. It induces a wide class of predictors whose evaluation involves quadratic forms of smooth functions of the unknown covariance. Our proposed adaptive prediction procedure outperforms factor model based plug-in predictors by using uniformly consistent estimators of the quadratic forms involved in the coordinate-wise shrinkage strategies. We further improve our predictor by introducing possible reduction in its variability through a novel coordinate-wise shrinkage policy that only uses covariance level information and can be adaptively tuned using the sample eigen structure of the high dimensional spiked covariance model. Simulation studies are conducted to show that in many settings the proposed method substantially improves the performance of traditional plug-in based shrinkage procedures which first estimate the covariance and thereafter optimize over the hyper-parameters.