加权残差经验过程、鞅变换与参数个数发散的回归模型设定检验
Weighted residual empirical processes, martingale transformations, and model specification tests for regressions with diverging number of parameters
AI总结：
本文针对参数个数发散的回归模型，提出基于加权残差经验过程的设定检验，无鞅变换检验具有最优局部备择检测速率且不依赖维度，虽非分布无关但可用平滑残差自助逼近，而鞅变换检验虽分布无关但功效较低。
摘要：
本文探讨了在发散维度设置下，回归模型中均值函数和方差函数的参数形式的假设检验问题。为缓解维数灾难，我们引入了基于加权残差经验过程的检验，包括有无鞅变换两种形式。这些检验的渐近性质由加权残差经验过程及其鞅变换在原假设和备择假设下的行为推导得出。未采用鞅变换的所提检验能够以最快可能的速率检测局部备择假设，具体为$n^{-1/2}$阶，该速率不受维度影响。然而，这些检验并非渐近分布无关的。为解决此局限性，我们提出了一种平滑残差自助逼近方法，并验证了其在发散维度设置下的有效性。相比之下，包含鞅变换的检验是渐近分布无关的，但表现出一个意外的局限性：它们只能检测以慢得多的速率（$n^{-1/4}$阶）收敛于原假设的局部备择假设，该速率同样独立于维度。这一发现揭示了基于无鞅变换的加权残差经验过程的检验在功效上相对于基于鞅变换的对应检验具有理论优势，挑战了现有基于鞅变换的渐近分布无关检验的传统观念。为验证我们的方法，我们进行了模拟研究，并将所提检验应用于一个真实数据集，证明了其实际有效性。
英文摘要：
This paper explores hypothesis testing for the parametric forms of the mean and variance functions in regression models under diverging-dimension settings. To mitigate the curse of dimensionality, we introduce weighted residual empirical process-based tests, both with and without martingale transformations. The asymptotic properties of these tests are derived from the behavior of weighted residual empirical processes and their martingale transformations under the null and alternative hypotheses. The proposed tests without martingale transformations achieve the fastest possible rate of detecting local alternatives, specifically of order $n^{-1/2}$, which is unaffected by dimensionality. However, these tests are not asymptotically distribution-free. To address this limitation, we propose a smooth residual bootstrap approximation and establish its validity in diverging-dimension settings. In contrast, tests incorporating martingale transformations are asymptotically distribution-free but exhibit an unexpected limitation: they can only detect local alternatives converging to the null at a much slower rate of order $n^{-1/4}$, which remains independent of dimensionality. This finding reveals a theoretical advantage in the power of tests based on weighted residual empirical process without martingale transformations over their martingale-transformed counterparts, challenging the conventional wisdom of existing asymptotically distribution-free tests based on martingale transformations. To validate our approach, we conduct simulation studies and apply the proposed tests to a real-world dataset, demonstrating their practical effectiveness.
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