Generalized Fiducial Inference for Ultrahigh-Dimensional Regression

In recent years, the ultrahigh-dimensional linear regression problem has attracted enormous attention from the research community. Under the sparsity assumption, most of the published work is devoted to the selection and estimation of the predictor variables with nonzero coefficients. This article studies a different but fundamentally important aspect of this problem: uncertainty quantification for parameter estimates and model choices. To be more specific, this article 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 correct asymptotic frequentist property. In terms of empirical performance, the proposed methods are tested by simulation experiments and an application to a real dataset. Finally, 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. A connection to objective Bayesian model selection is also discussed.