  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.
