Nonparametric regression models do not require the specification of the functional form between the outcome and the covariates. Despite their popularity, the amount of diagnostic statistics, in comparison to their parametric counter-parts, is small. We propose a goodness-of-fit test for nonparametric regression models with linear smoother form. In particular, we apply this testing framework to smoothing spline ANOVA models. The test can consider two sources of lack-of-fit: whether covariates that are not currently in the model need to be included, and whether the current model fits the data well. The proposed method derives estimated residuals from the model. Then, statistical dependence is assessed between the estimated residuals and the covariates using the HSIC. If dependence exists, the model does not capture all the variability in the outcome associated with the covariates, otherwise the model fits the data well. The bootstrap is used to obtain p-values. Application of the method is demonstrated with a neonatal mental development data analysis. We demonstrate correct type I error as well as power performance through simulations.

中文翻译：

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非参数回归模型的拟合优度检验：以平滑样条方差分析模型为例

非参数回归模型不需要指定结果和协变量之间的函数形式。尽管它们很受欢迎，但与它们的参数对应部分相比，诊断统计的数量很少。我们为具有线性平滑形式的非参数回归模型提出了一种拟合优度检验。特别是，我们将此测试框架应用于平滑样条方差分析模型。检验可以考虑失拟的两个来源：是否需要包括当前模型中没有的协变量，以及当前模型是否很好地拟合了数据。所提出的方法从模型中推导出估计的残差。然后，使用 HSIC 评估估计残差和协变量之间的统计相关性。如果存在依赖，该模型不会捕获与协变量相关的结果中的所有可变性，否则该模型会很好地拟合数据。引导程序用于获得 p 值。通过新生儿心理发育数据分析证明了该方法的应用。我们通过模拟演示了正确的 I 类错误以及功率性能。