This paper studies model checking for general parametric regression models having no dimension reduction structures on the predictor vector. Using any U-statistic type test as an initial test, this paper combines the sample-splitting and conditional studentization approaches to construct a COnditionally Studentized Test (COST). Whether the initial test is global or local smoothing-based; the dimension of the predictor vector and the number of parameters are fixed or diverge at a certain rate, the proposed test always has a normal weak limit under the null hypothesis. When the dimension of the predictor vector diverges to infinity at faster rate than the number of parameters, even the sample size, these results are still available under some conditions. This shows the potential of our method to handle higher dimensional problems. Further, the test can detect the local alternatives distinct from the null hypothesis at the fastest possible rate of convergence in hypothesis testing. We also discuss the optimal sample splitting in power performance. The numerical studies offer information on its merits and limitations in finite sample cases including the setting where the dimension of predictor vector equals the sample size. As a generic methodology, it could be applied to other testing problems.

翻译：本文研究预测向量不具有降维结构的一般参数回归模型的模型检验问题。以任意U统计量型检验为初始检验，本文结合样本分割与条件学生化方法，构建了条件学生化检验（COST）。无论初始检验是基于全局平滑还是局部平滑方法，无论预测向量维度和参数数目是固定还是以特定速率发散，所提出的检验在原假设下始终具有正态弱极限。当预测向量维度以快于参数数目的速率发散至无穷大，甚至超过样本量时，该结果在特定条件下仍然成立，这表明了本方法处理更高维问题的潜力。此外，该检验能够以假设检验中最优的收敛速率检测偏离原假设的局部备择假设。我们还讨论了检验功效角度下的最优样本分割策略。数值研究展示了该方法在有限样本情形（包括预测向量维度等于样本量的设定）中的优势与局限。作为一种通用方法论，该方法可应用于其他检验问题。