Generalized Fiducial Inference for Ultrahigh Dimensional Regression

TL;DR

This paper introduces a novel approach using generalized fiducial inference to quantify uncertainty in ultrahigh dimensional linear regression, providing confidence intervals and model probabilities with strong theoretical guarantees.

Contribution

It applies the generalized fiducial methodology to high-dimensional regression, offering the first such application to 'large p small n' problems with proven asymptotic properties.

Findings

Methods have exact asymptotic frequentist properties.

Simulation experiments demonstrate good empirical performance.

Application to real data showcases practical utility.

Abstract

In recent years the ultrahigh dimensional linear regression problem has attracted enormous attentions from the research community. Under the sparsity assumption most of the published work is devoted to the selection and estimation of the significant predictor variables. This paper studies a different but fundamentally important aspect of this problem: uncertainty quantification for parameter estimates and model choices. To be more specific, this paper proposes methods for deriving a probability density function on the set of all possible models, and also for constructing confidence intervals for the corresponding parameters. These proposed methods are developed using the generalized fiducial methodology, which is a variant of Fisher's controversial fiducial idea. Theoretical properties of the proposed methods are studied, and in particular it is shown that statistical inference based on the proposed methods will have exact asymptotic frequentist property. In terms of empirical performances, the proposed methods are tested by simulation experiments and an application to a real data set. Lastly this work can also be seen as an interesting and successful application of Fisher's fiducial idea to an important and contemporary problem. To the best of the authors' knowledge, this is the first time that the fiducial idea is being applied to a so-called "large p small n" problem.