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2020 Estimation of linear projections of non-sparse coefficients in high-dimensional regression
David Azriel, Armin Schwartzman
Electron. J. Statist. 14(1): 174-206 (2020). DOI: 10.1214/19-EJS1656

Abstract

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.

Citation

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David Azriel. Armin Schwartzman. "Estimation of linear projections of non-sparse coefficients in high-dimensional regression." Electron. J. Statist. 14 (1) 174 - 206, 2020. https://doi.org/10.1214/19-EJS1656

Information

Received: 1 February 2019; Published: 2020
First available in Project Euclid: 7 January 2020

zbMATH: 07154986
MathSciNet: MR4047998
Digital Object Identifier: 10.1214/19-EJS1656

Subjects:
Primary: 60K35 , 62J05
Secondary: 62P10

Keywords: high-dimensional regression , linear projections

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Vol.14 • No. 1 • 2020
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