  This article studies the derivatives in models that flexibly characterize the
relationship between a response variable and multiple predictors, with goals of
providing both accurate estimation and inference procedures for hypothesis
testing. In the setting of tensor product reproducing spaces for nonparametric
multivariate functions, we propose a plug-in kernel ridge regression estimator
to estimate the derivatives of the underlying multivariate regression function
under 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-depth
analysis establishing the minimax lower bound, which matches 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 the 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 methods.
