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June 2020 Large sample properties of partitioning-based series estimators
Matias D. Cattaneo, Max H. Farrell, Yingjie Feng
Ann. Statist. 48(3): 1718-1741 (June 2020). DOI: 10.1214/19-AOS1865


We present large sample results for partitioning-based least squares nonparametric regression, a popular method for approximating conditional expectation functions in statistics, econometrics and machine learning. First, we obtain a general characterization of their leading asymptotic bias. Second, we establish integrated mean squared error approximations for the point estimator and propose feasible tuning parameter selection. Third, we develop pointwise inference methods based on undersmoothing and robust bias correction. Fourth, employing different coupling approaches, we develop uniform distributional approximations for the undersmoothed and robust bias-corrected $t$-statistic processes and construct valid confidence bands. In the univariate case, our uniform distributional approximations require seemingly minimal rate restrictions and improve on approximation rates known in the literature. Finally, we apply our general results to three partitioning-based estimators: splines, wavelets and piecewise polynomials. The Supplemental Appendix includes several other general and example-specific technical and methodological results. A companion $\mathsf{R}$ package is provided.


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Matias D. Cattaneo. Max H. Farrell. Yingjie Feng. "Large sample properties of partitioning-based series estimators." Ann. Statist. 48 (3) 1718 - 1741, June 2020.


Received: 1 November 2018; Revised: 1 May 2019; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241609
MathSciNet: MR4124341
Digital Object Identifier: 10.1214/19-AOS1865

Primary: 62E20 , 62G08 , 62G20
Secondary: 62M99

Keywords: Nonparametric regression , robust bias correction , series methods , sieve methods , strong approximation , tuning parameter selection , uniform inference

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 3 • June 2020
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