文章基本信息

标题：Estimation of linear projections of non-sparse coefficients in high-dimensional regression
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作者：David Azriel ; Armin Schwartzman
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2020
卷号：14
期号：1
页码：174-206
DOI：10.1214/19-EJS1656
语种：English
出版社：Institute of Mathematical Statistics
摘要：In this work we study estimation of signals when the number of parameters is much larger than the number of observations. A large body of literature assumes for these kind of problems a sparse structure where most of the parameters are zero or close to zero. When this assumption does not hold, one can focus on low-dimensional functions of the parameter vector. In this work we study one-dimensional linear projections. Specifically, in the context of high-dimensional linear regression, the parameter of interest is ${\boldsymbol{\beta}}$ and we study estimation of $\mathbf{a}^{T}{\boldsymbol{\beta}}$. We show that $\mathbf{a}^{T}\hat{\boldsymbol{\beta}}$, where $\hat{\boldsymbol{\beta}}$ is the least squares estimator, using pseudo-inverse when $p>n$, is minimax and admissible. Thus, for linear projections no regularization or shrinkage is needed. This estimator is easy to analyze and confidence intervals can be constructed. We study a high-dimensional dataset from brain imaging where it is shown that the signal is weak, non-sparse and significantly different from zero.
关键词：High-dimensional regression; linear projections