The lower tail of random quadratic forms with applications to ordinary least squares

• 成果类型: Article
• 署名作者: Oliveira, Roberto Imbuzeiro
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0738-9
• 发表日期: 2016
• 页码: 1175-1194
• 关键词: lower bounds; matrices

摘要

Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the lower tail of such a matrix, and prove that it is sub-Gaussian under a simple fourth moment assumption on the one-dimensional marginals of the random vectors. A similar result holds for more general sums of random positive semidefinite matrices, and our (relatively simple) proof uses a variant of the so-called PAC-Bayesian method for bounding empirical processes. Using this bound, we obtain a nearly optimal finite-sample result for the ordinary least squares estimator under random design.