Uncertainty Quantification for High-Dimensional Sparse Nonparametric Additive Models

Abstract

Statistical inference in high dimensional settings has recently attracted\nenormous attention within the literature. However, most published work focuses\non the parametric linear regression problem. This paper considers an important\nextension of this problem: statistical inference for high dimensional sparse\nnonparametric additive models. To be more precise, this paper develops a\nmethodology for constructing a probability density function on the set of all\ncandidate models. This methodology can also be applied to construct confidence\nintervals for various quantities of interest (such as noise variance) and\nconfidence bands for the additive functions. This methodology is derived using\na generalized fiducial inference framework. It is shown that results produced\nby the proposed methodology enjoy correct asymptotic frequentist properties.\nEmpirical results obtained from numerical experimentation verify this\ntheoretical claim. Lastly, the methodology is applied to a gene expression data\nset and discovers new findings for which most existing methods based on\nparametric linear modeling failed to observe.\n