Large sample properties of partitioning-based series estimators

Matias D. Cattaneo, Max H. Farrell, Yingjie Feng

Research output: Contribution to journalArticle

Abstract

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 R package is provided.

Original languageEnglish (US)
Pages (from-to)1718-1741
Number of pages24
JournalAnnals of Statistics
Volume48
Issue number3
DOIs
StatePublished - Jun 2020

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Nonparametric regression
  • Robust bias correction
  • Series methods
  • Sieve methods
  • Strong approximation
  • Tuning parameter selection
  • Uniform inference

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