In multivariate nonparametric regression the additive models are very useful when a suitable parametric model is difficult to find. The backfitting algorithm is a powerful tool to estimate the additive components. However, due to complexity of the estimators, the asymptotic $p$-value of the associated test is difficult to calculate without a Monte Carlo simulation. Moreover, the conventional tests assume that the predictor variables are strictly continuous. In this paper, a new test is introduced for the additive components with discrete or categorical predictors, where the model may contain continuous covariates. This method is also applied to the semiparametric regression to test the goodness-of-fit of the model. These tests are asymptotically optimal in terms of the rate of convergence, as they can detect a specific class of contiguous alternatives at a rate of $n^{-1/2}$. An extensive simulation study is presented to support the theoretical results derived in this paper. Finally, the method is applied to a real data to model the diamond price based on its quality attributes and physical measurements.

中文翻译：

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具有离散或分类预测变量的加性模型的最佳检验

在多元非参数回归中，当难以找到合适的参数模型时，加性模型非常有用。反向拟合算法是估计加性分量的强大工具。然而，由于估计器的复杂性，如果没有蒙特卡罗模拟，则很难计算相关测试的渐近 $p$ 值。此外，传统测试假设预测变量是严格连续的。在本文中，针对具有离散或分类预测变量的加性分量引入了一种新测试，其中模型可能包含连续协变量。这种方法也适用于半参数回归来检验模型的拟合优度。这些测试在收敛速度方面是渐近最优的，因为他们可以以 $n^{-1/2}$ 的速度检测特定类别的连续替代品。提供了广泛的模拟研究来支持本文得出的理论结果。最后，将该方法应用于真实数据，以根据钻石的质量属性和物理测量值对钻石价格进行建模。