The integrated conditional moment (ICM) test is a classical and widely used method for assessing the adequacy of regression models. Although it performs well in fixed-dimension settings, its behavior changes dramatically when the predictor dimension diverges: in such regimes, the limiting null and alternative distributions of the ICM statistic degenerate to fixed constants. Moreover, when the number of predictors diverges, the commonly used wild bootstrap no longer approximates the null distribution of the ICM statistic well, leading to size distortion and substantial power loss. To address these challenges, we propose a new specification test based on weighted residual processes for evaluating the parametric form of the regression mean function in high-dimensional settings where the number of predictors increases with the sample size. We establish the asymptotic properties of the test statistic under the null hypothesis and under global and local alternatives. The proposed test maintains the nominal significance level and can detect local alternatives that deviate from the null hypothesis at the parametric rate $1/\sqrt{n}$. Furthermore, we propose a smooth residual bootstrap to approximate the limiting null distribution and establish its validity in high-dimensional settings. Two simulation studies and a real-data example are conducted to evaluate the finite-sample performance of the proposed test.

翻译：综合条件矩(ICM)检验是评估回归模型适用性的经典且广泛使用的方法。虽然在固定维度设置下表现良好，但当预测变量维度发散时，其行为会发生显著变化：在此类情况下，ICM统计量的极限原假设分布与备择分布退化为固定常数。此外，当预测变量数量发散时，常用的wild bootstrap方法不再能良好近似ICM统计量的原假设分布，导致检验水平扭曲及功效显著下降。针对这些挑战，本文提出一种基于加权残差过程的新型设定检验方法，用于在预测变量数量随样本量增长的高维场景下评估回归均值函数的参数形式。我们建立了该检验统计量在原假设、全局备择以及局部备择假设下的渐近性质。所提检验能够维持名义显著性水平，并可检测以参数速率$1/\sqrt{n}$偏离原假设的局部备择假设。此外，我们提出一种光滑残差bootstrap方法来近似极限原假设分布，并证明其在维数发散场景中的有效性。通过两项模拟研究及一项实际数据案例，验证了所提检验在有限样本下的表现。