Confidence Intervals for Low-Dimensional Parameters in High-Dimensional Linear Models

Abstract

The purpose of this paper is to propose methodologies for statistical inference of low-dimensional parameters with high-dimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broad context. The theoretical results presented here provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finite-dimensional covariance matrices. These sufficient conditions allow the number of variables to far exceed the sample size. The simulation results presented here demonstrate the accuracy of the coverage probability of the proposed confidence intervals, strongly supporting the theoretical results.

References in corpus (3)

• Nearly unbiased variable selection under minimax concave penalty
• The Dantzig selector and sparsity oracle inequalities
• A General Theory of Concave Regularization for High Dimensional Sparse Estimation Problems

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• Group-bound: confidence intervals for groups of variables in sparse high-dimensional regression without assumptions on the design
• Gaussian Graphical Model Estimation with False Discovery Rate Control
• A Global Homogeneity Test for High-Dimensional Linear Regression