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This article studies the derivatives in models that flexibly characterize therelationship between a response variable and multiple predictors, with goals ofproviding both accurate estimation and inference procedures for hypothesistesting. In the setting of tensor product reproducing spaces for nonparametricmultivariate functions, we propose a plug-in kernel ridge regression estimatorto estimate the derivatives of the underlying multivariate regression functionunder the smoothing spline ANOVA model. This estimator has an analytical form,making it simple to implement in practice. We first establish $L_\infty$ and$L_2$ convergence rates of the proposed estimator under general random designs.For derivatives with some selected interesting orders, we provide an in-depthanalysis establishing the minimax lower bound, which matches the $L_2$convergence rate. Additionally, motivated by a wide range of applications, wepropose a hypothesis testing procedure to examine whether a derivative is zero.Theoretical results demonstrate that the proposed testing procedure achievesthe correct size under the null hypothesis and is asymptotically powerful underlocal alternatives. For ease of use, we also develop an associated bootstrapalgorithm to construct the rejection region and calculate the p-value, and theconsistency of the proposed algorithm is established. Simulation studies usingsynthetic data and an application to a real-world dataset confirm theeffectiveness of our methods.

Estimation and Hypothesis Testing of Derivatives in Smoothing Spline ANOVA Models