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

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• 作者：Roberto Imbuzeiro Oliveira
• 关键词：Random covariance matrices ; Linear regression
• 刊名：Probability Theory and Related Fields
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：166
• 期：3-4
• 页码：1175-1194
• 全文大小：503 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Mathematical and Computational Physics
  Quantitative Finance
  Mathematical Biology
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Operation Research and Decision Theory
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-2064
• 卷排序：166

文摘

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.