Ultrahigh‐dimensional generalized additive model: Unified theory and methods

Abstract

AbstractGeneralized additive model is a powerful statistical learning and predictive modeling tool that has been applied in a wide range of applications. The need of high‐dimensional additive modeling is eminent in the context of dealing with high throughput data such as genetics data analysis. In this article, we studied a two‐step selection and estimation method for ultrahigh‐dimensional generalized additive models. The first step applies group lasso on the expanded bases of the functions. With high probability this selects all nonzero functions without having too much over selection. The second step uses adaptive group lasso with any initial estimators, including the group lasso estimator, that satisfies some regular conditions. The adaptive group lasso estimator is shown to be selection consistent with improved convergence rates. Tuning parameter selection is also discussed and shown to select the true model consistently under generalized information criterion procedure. The theoretical properties are supported by extensive numerical study.