Traditional nonparametric estimation methods often lead to a slow convergence rate in large dimensions and require unrealistically large dataset sizes for reliable conclusions. We develop an approach based on partial derivatives, either observed or estimated, to effectively estimate the function at near-parametric convergence rates. This novel approach and computational algorithm could lead to methods useful to practitioners in many areas of science and engineering. Our theoretical results reveal behaviour universal to this class of nonparametric estimation problems. We explore a general setting involving tensor product spaces and build upon the smoothing spline analysis of variance framework. For d-dimensional models under full interaction, the optimal rates with gradient information on p covariates are identical to those for the $(d-p)$ -interaction models without gradients and, therefore, the models are immune to the curse of interaction. For additive models, the optimal rates using gradient information are $\sqrt{n}$ , thus achieving the parametric rate. We demonstrate aspects of the theoretical results through synthetic and real data applications.

Keywords: derivatives; interactions; rates of convergence; reproducing kernel Hilbert space; smoothing spline ANOVA.

© The Royal Statistical Society 2024.

传统的非参数估计方法在高维情况下通常导致收敛速度缓慢，并且需要不切实际的大数据集才能得出可靠结论。我们开发了一种基于偏导数（无论是观察到的还是估计出的）的方法，以接近参数级的收敛速率有效估计函数。这种新颖的方法和计算算法可能对科学与工程领域的从业者非常有用。我们的理论结果揭示了这类非参数估计问题中的普遍行为。我们在涉及张量积空间的一般设定中进行探讨，并在光滑样条分析方差框架上进行了研究。对于具有完整交互作用的d维模型，在p个协变量上的梯度信息下的最优速率与没有梯度信息的$(d\; -\; p)$-交互模型相同，因此这些模型不受“交互诅咒”的影响。对于加性模型，使用梯度信息的最佳收敛率为$sqrt\{n\}$，从而达到了参数级的收敛率。我们通过合成数据和实际数据分析展示了理论结果的一些方面。

关键词： 导数；交互作用；收敛速度；再生希尔伯特空间；光滑样条ANOVA。

© The Royal Statistical Society 2024.