We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, \textbf{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) and a loss function (LF) 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.

翻译：我们调查非参数添加模型的假设测试,这些模型使用简化的平整调整估计(Huang和Yu,《计算和图形统计杂志》,\ textbf{28(2)}28(2)},386-400;2019); 简化的平整回配在正常条件下实现或骨骼特性,并提供有助于得出无光化特性的定点数的封闭式表达方式; 我们开发了普遍概率比(GLR)和基于损失的测试框架,以推断。 在无效假设下,GLR和LF的测试都具有无源重标的奇质配方分布,而且两者都展示了威尔克斯现象,这意味着伸缩常数和自由度与扰动参数无关。这些测试在非参数测算的趋同率方面几乎是最佳的。此外,适合模型估计的带宽度功能可能有益于测试。在添加模型中,我们显示LF的测试比GLR测试和G的实用性模型要强大得多。我们用模拟了这些GLR的模型来说明其实际功率测试。