Estimation and Hypothesis Testing of Derivatives in Smoothing Spline ANOVA Models

TL;DR

This paper introduces a new kernel ridge regression estimator for derivatives in smoothing spline ANOVA models, providing convergence rates, hypothesis testing procedures, and bootstrap methods validated through simulations and real data.

Contribution

It develops a plug-in estimator for derivatives, establishes convergence rates, and proposes a hypothesis testing framework with bootstrap algorithms within smoothing spline ANOVA models.

Findings

Estimator achieves optimal $L_ extinfty$ convergence rate.

Hypothesis test correctly controls size and is powerful.

Bootstrap algorithm is consistent and effective.

Abstract

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.