Uncertainty Quantification for High-Dimensional Sparse Nonparametric Additive Models

Abstract

Statistical inference in high-dimensional settings has recently attracted enormous attention within the literature. However, most published work focuses on the parametric linear regression problem. This article considers an important extension of this problem: statistical inference for high-dimensional sparse nonparametric additive models. To be more precise, this article develops a methodology for constructing a probability density function on the set of all candidate models. This methodology can also be applied to construct confidence intervals for various quantities of interest (such as noise variance) and confidence bands for the additive functions. This methodology is derived using a generalized fiducial inference framework. It is shown that results produced by the proposed methodology enjoy correct asymptotic frequentist properties. Empirical results obtained from numerical experimentation verify this theoretical claim. Lastly, the methodology is applied to a gene expression dataset and discovers new findings for which most existing methods based on parametric linear modeling failed to observe.