Functional ANOVA provides a nonparametric modeling framework for multivariate covariates, enabling flexible estimation and interpretation of effect functions such as main effects and interaction effects. However, effect-wise inference in such models remains challenging. Existing methods focus primarily on inference for entire functions rather than individual effects. Methods addressing effect-wise inference face substantial limitations: the inability to accommodate interactions, a lack of rigorous theoretical foundations, or restriction to pointwise inference. To address these limitations, we develop a unified framework for effect-wise inference in smoothing spline ANOVA on a subspace of tensor product Sobolev space. For each effect function, we establish rates of convergence, pointwise confidence intervals, and a Wald-type test for whether the effect is zero, with power achieving the minimax distinguishable rate up to a logarithmic factor. Main effects achieve the optimal univariate rates, and interactions achieve optimal rates up to logarithmic factors. The theoretical foundation relies on an orthogonality decomposition of effect subspaces, which enables the extension of the functional Bahadur representation framework to effect-wise inference in smoothing spline ANOVA with interactions. Simulation studies and real-data application to the Colorado temperature dataset demonstrate superior performance compared to existing methods.

翻译：函数方差分析为多变量协变量提供了一种非参数建模框架，能够灵活估计和解释主效应、交互效应等效果函数。然而，此类模型中的效果推断仍然具有挑战性。现有方法主要关注整体函数的推断而非单个效应。针对效果推断的方法面临显著局限：无法处理交互效应、缺乏严格理论基础，或仅限于逐点推断。为突破这些局限，我们针对张量积Sobolev空间子空间上的光滑样条方差分析，开发了一个统一的效果推断框架。对于每个效应函数，我们建立了收敛速率、逐点置信区间，以及关于效应是否为零的Wald型检验，其检验功效在忽略对数因子的情况下达到极小可区分速率。主效应达到最优单变量速率，交互效应在忽略对数因子的情况下达到最优速率。该理论基础依赖于效应子空间的正交分解，这使得函数Bahadur表示框架能够扩展至含交互效应的光滑样条方差分析的效果推断。基于模拟研究与科罗拉多温度数据集的实证应用表明，该方法相较于现有方法具有更优性能。