On Degrees of Freedom of Projection Estimators With Applications to Multivariate Nonparametric Regression. Issue 529 (2nd January 2020)

• Record Type: Journal Article
• Authors: Chen, Xi; Lin, Qihang; Sen, Bodhisattva
• Is Part Of: Journal of the American Statistical Association. Volume 115:Issue 529(2020)
• Page Start: 173
• Page End: 186
• Publication Date: 2020-01-02
• Subjects: Additive model -- Bounded isotonic regression -- Divergence of an estimator -- Generalized Lasso -- Multivariate convex regression; Statistics -- Periodicals; Statistics -- Periodicals; Statistiques -- Périodiques; États-Unis -- Statistiques -- Périodiques; 519.5
• DOI: 10.1080/01621459.2018.1537917
• Languages: English
• ISSNs: 0162-1459

Abstract

Abstract : Abstract– In this article, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as outputs of linearly constrained quadratic optimization procedures; namely, minimizers of the least-squares criterion with linear constraints and/or quadratic penalties. As special cases of our results, we derive explicit expressions for the degrees of freedom in many nonparametric regression problems, for example, bounded isotonic regression, multivariate (penalized) convex regression, and additive total variation regularization. Our theory also yields, as special cases, known results on the degrees of freedom of many well-studied estimators in the statistics literature, such as ridge regression, Lasso and generalized Lasso. Our results can be readily used to choose the tuning parameter(s) involved in the estimation procedure by minimizing the Stein's unbiased risk estimate. As a by-product of our analysis we derive an interesting connection between bounded isotonic regression and isotonic regression on a general partially ordered set, which is of independent interest. Supplementary materials for this article are available online.