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.
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 develop 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 popular partition-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.
R
employing a carefully crafted coupling approach
we obtain a general characterization of their leading asymptotic bias. Second, we establish integrated mean squared error approximations for the point estimator and develop feasible tuning parameter selection. Third, we develop pointwise inference methods based on undersmoothing and robust bias correction. Fourth, employing different coupling approaches
A
the undersmoothed and robust bias corrected
R
processes (indexed over the support of the conditioning variables). These results cover both undersmoothed and bias corrected inference,
processes and construct valid confidence bands. In the univariate case, our uniform distributional approximations
R
, 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
and improve on approximation rates known in the literature. Finally, we
D
(generalizing the regressogram)
A
several
R
results that may be of independent interest
and methodological results