Moderate-Dimensional Inferences on Quadratic Functionals in Ordinary Least Squares

• 成果类型: Article
• 署名作者: Guo, Xiao; Cheng, Guang
• 署名单位: Chinese Academy of Sciences; University of Science & Technology of China, CAS; Purdue University System; Purdue University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2021.1893177
• 发表日期: 2022
• 页码: 1931-1950
• 关键词: p-regression parameters; confidence-intervals; asymptotic-behavior; M-ESTIMATORS; p2/n

摘要

Statistical inferences for quadratic functionals of linear regression parameter have found wide applications including signal detection, global testing, inferences of error variance and fraction of variance explained. Classical theory based on ordinary least squares estimator works perfectly in the low-dimensional regime, but fails when the parameter dimension pn grows proportionally to the sample size n. In some cases, its performance is not satisfactory even when n >= 5p(n). The main contribution of this article is to develop dimension-adaptive inferences for quadratic functionals when lim(n ->infinity) p(n)/n = tau is an element of [0, 1). We propose a bias-and-variance-corrected test statistic and demonstrate that its theoretical validity (such as consistency and asymptotic normality) is adaptive to both low dimension with tau = 0 and moderate dimension with tau is an element of (0, 1). Our general theory holds, in particular, without Gaussian design/error or structural parameter assumption, and applies to a broad class of quadratic functionals covering all aforementioned applications. As a by-product, we find that the classical fixed-dimensional results continue to hold if and only if the signal-to-noise ratio is large enough, say when pn diverges but slower than n. Extensive numerical results demonstrate the satisfactory performance of the proposed methodology even when pn = 0.9n in some extreme cases. The mathematical arguments are based on the random matrix theory and leave-oneobservation- out method.