We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, 28(2), 386–400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) (Fan, Zhang and Zhang, Annals of statistics, 29(1),153–193, 2001) and a loss function (LF) (Hong and Lee, Annals of Statistics, 41(3), 1166–1203, 2013)-based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled chi-squared distributions, and both exhibit the Wilks phenomenon, which means the scaling constants and degrees of freedom are independent of nuisance parameters. These tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing. Additionally, the bandwidths that are well suited for model estimation may be useful for testing. We show that in additive models, the LF test is asymptotically more powerful than the GLR test. We use simulations to demonstrate the Wilks phenomenon and the power of these proposed GLR and LF tests, and a real example to illustrate their usefulness.

我们研究了使用简化平滑反向拟合估计的非参数加法模型中的假设检验（Huang 和 Yu，Journal of Computational and Graphical Statistics，28(2)，386–400，2019）。简化的平滑反向拟合在正则条件下实现了预言属性，并提供了可用于推导渐近属性的估计量的封闭形式表达式。我们开发了一个广义似然比 (GLR) (Fan, Zhang and Zhang, Annals of Statistics, 29(1),153–193, 2001) 和一个损失函数 (LF) (Hong and Lee, Annals of Statistics, 41(3 ), 1166–1203, 2013) 基于测试框架的推理。在原假设下，GLR 和 LF 检验都具有渐近重新缩放的卡方分布，并且都表现出 Wilks 现象，这意味着缩放常数和自由度与讨厌的参数无关。这些检验在非参数假设检验的收敛速度方面是渐近最优的。此外，非常适合模型估计的带宽可能对测试有用。我们表明，在加法模型中，LF 检验比 GLR 检验更强大。我们使用模拟来展示 Wilks 现象以及这些提议的 GLR 和 LF 测试的威力，并用一个真实的例子来说明它们的有用性。LF 测试在渐近上比 GLR 测试更强大。我们使用模拟来展示 Wilks 现象以及这些提议的 GLR 和 LF 测试的威力，并用一个真实的例子来说明它们的有用性。LF 测试在渐近上比 GLR 测试更强大。我们使用模拟来展示 Wilks 现象以及这些提议的 GLR 和 LF 测试的威力，并用一个真实的例子来说明它们的有用性。