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：函数方差分析为多元协变量提供了非参数建模框架，能够灵活估计和解释主效应及交互效应等效应函数。然而，此类模型中的效应推断仍面临挑战。现有方法主要关注整体函数的推断而非个体效应。针对效应推断的方法存在显著局限性：无法处理交互效应、缺乏严格理论基础，或仅限于逐点推断。为解决这些局限，我们在张量积Sobolev空间的子空间上，为平滑样条方差分析开发了统一的效应推断框架。针对每个效应函数，我们建立了收敛速率、逐点置信区间，以及检验效应是否为零的Wald型检验，其检验功效在达到对数因子范围内实现了极小极大可区分速率。主效应达到最优单变量速率，交互效应在达到对数因子范围内实现最优速率。该理论基础依赖于效应子空间的正交分解，从而将函数Bahadur表示框架扩展至包含交互效应的平滑样条方差分析中的效应推断。通过模拟研究和科罗拉多温度数据集的实际应用，本方法相较于现有方法展现出更优越的性能。