Within the framework of smoothing spline ANOVA, we propose a plug-in kernel ridge regression estimator to estimate the derivatives of the underlying multivariate regression function. We first establish an $L_\infty$ convergence rate of the proposed estimator under general random designs. When the covariates are uniformly distributed, we provide a in-depth analysis that includes a sharp upper bound and the minimax lower bound of the $L_2$ convergence rate. Additionally, motivated by a wide range of applications, we propose a hypothesis testing procedure to examine whether a derivative is zero. Theoretical results demonstrate that the proposed testing procedure achieves the correct size under the null hypothesis and is asymptotically powerful under local alternatives. For ease of use, we also develop an associated bootstrap algorithm to construct the rejection region and calculate p-value, and the consistency of the proposed algorithm is established. Simulation studies using synthetic data and an application to a real-world dataset confirm the effectiveness of our approach.

翻译：在光滑样条方差分析框架下，我们提出了一种插件式核岭回归估计量，用于估计潜在多元回归函数的导数。我们首先在一般随机设计下建立了所提估计量的$L_\infty$收敛速率。当协变量服从均匀分布时，我们进行了深入分析，给出了$L_2$收敛速率的尖锐上界和极小极大下界。此外，受广泛应用的启发，我们提出了一种假设检验程序来检验导数是否为零。理论结果表明，所提检验程序在零假设下具有正确的检验水平，并在局部备择假设下具有渐近功效。为便于使用，我们还开发了相应的自助算法以构建拒绝域并计算p值，并证明了该算法的一致性。通过合成数据的模拟研究及对真实数据集的实证应用，验证了我们方法的有效性。