Large Sample Properties of Partitioning-Based Series Estimators
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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, employing a carefully crafted coupling approach, we develop uniform distributional approximations for t-statistic processes (indexed over the support of the conditioning variables). These results cover both undersmoothed and bias corrected inference, require seemingly minimal rate restrictions, and achieve fast approximation rates. Using the uniform approximations we construct valid confidence bands which share these same advantages. While our coupling approach exploits specific features of the estimators considered, the underlying ideas may be useful in other contexts more generally. Second, we obtain valid integrated mean square error approximations for partitioning-based estimators and develop feasible tuning parameter selection. We apply our general results to three popular partition-based estimators: splines, wavelets, and piecewise polynomials (generalizing the regressogram). The supplemental appendix includes other general and example-specific technical results that may be of independent interest. A companion R package is provided.
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