Estimation in high-dimensional linear models with deterministic design matrices

TL;DR

This paper addresses high-dimensional linear models with deterministic design matrices, proposing a ridge regression-based estimator for the projection of the parameter vector, with proven asymptotic properties and demonstrated through simulations.

Contribution

It introduces a novel thresholded ridge regression estimator for sparse projection vectors in high-dimensional deterministic design models, with explicit form and theoretical guarantees.

Findings

Estimator is consistent for variable selection and estimation.

Convergence rate of prediction mean squared error is established.

Simulation results confirm the effectiveness of the proposed method.

Abstract

Because of the advance in technologies, modern statistical studies often encounter linear models with the number of explanatory variables much larger than the sample size. Estimation and variable selection in these high-dimensional problems with deterministic design points is very different from those in the case of random covariates, due to the identifiability of the high-dimensional regression parameter vector. We show that a reasonable approach is to focus on the projection of the regression parameter vector onto the linear space generated by the design matrix. In this work, we consider the ridge regression estimator of the projection vector and propose to threshold the ridge regression estimator when the projection vector is sparse in the sense that many of its components are small. The proposed estimator has an explicit form and is easy to use in application. Asymptotic properties such as the consistency of variable selection and estimation and the convergence rate of the prediction mean squared error are established under some sparsity conditions on the projection vector. A simulation study is also conducted to examine the performance of the proposed estimator.