## Electronic Journal of Statistics

### Thresholding least-squares inference in high-dimensional regression models

Mihai Giurcanu

#### Abstract

We propose a thresholding least-squares method of inference for high-dimensional regression models when the number of parameters, $p$, tends to infinity with the sample size, $n$. Extending the asymptotic behavior of the F-test in high dimensions, we establish the oracle property of the thresholding least-squares estimator when $p=o(n)$. We propose two automatic selection procedures for the thresholding parameter using Scheffé and Bonferroni methods. We show that, under additional regularity conditions, the results continue to hold even if $p=\exp(o(n))$. Lastly, we show that, if properly centered, the residual-bootstrap estimator of the distribution of thresholding least-squares estimator is consistent, while a naive bootstrap estimator is inconsistent. In an intensive simulation study, we assess the finite sample properties of the proposed methods for various sample sizes and model parameters. The analysis of a real world data set illustrates an application of the methods in practice.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 2 (2016), 2124-2156.

Dates
First available in Project Euclid: 18 July 2016

https://projecteuclid.org/euclid.ejs/1468849973

Digital Object Identifier
doi:10.1214/16-EJS1160

Mathematical Reviews number (MathSciNet)
MR3522671

Zentralblatt MATH identifier
1347.62131

Subjects
Primary: 62J05: Linear regression
Secondary: 62E20: Asymptotic distribution theory

#### Citation

Giurcanu, Mihai. Thresholding least-squares inference in high-dimensional regression models. Electron. J. Statist. 10 (2016), no. 2, 2124--2156. doi:10.1214/16-EJS1160. https://projecteuclid.org/euclid.ejs/1468849973

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