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.

中文翻译：

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基于分区的序列估计量的大样本属性

我们展示了基于分区的最小二乘非参数回归的大样本结果，这是一种在统计学、计量经济学和机器学习中逼近条件期望函数的流行方法。首先，我们获得了它们的主要渐近偏差的一般特征。其次，我们为点估计器建立了积分均方误差近似，并提出了可行的调整参数选择。第三，我们开发了基于欠平滑和稳健偏差校正的逐点推理方法。第四，采用不同的耦合方法，我们为欠平滑和稳健的偏差校正 t 统计过程开发了均匀分布近似，并构建了有效的置信带。在单变量情况下，我们的均匀分布近似需要看似最小的速率限制，并改进文献中已知的近似速率。最后，我们将我们的一般结果应用于三个基于分区的估计器：样条、小波和分段多项式。补充附录包括其他几个一般性和特定于示例的技术和方法学结果。提供了一个配套的 R 包。