基于投影经验过程和p值组合的超高维参数回归的渐近分布自由检验
Asymptotic Distribution-Free Tests for Ultra-high Dimensional Parametric Regressions via Projected Empirical Processes and $p$-value Combination
AI总结：
本文提出了一种基于投影经验过程和p值组合的混合检验方法，用于超高维参数回归模型的拟合优度检验，通过结合低频和高频信号检测提升检验效能。
摘要：
本文开发了一种新的方法，用于测试稀疏参数回归模型的拟合优度，基于投影经验过程和p值组合，其中协变量维度可能显著超过样本量。在这种超高维设置中，传统基于经验过程的检验往往失效，因为维度诅咒或其依赖于参数估计量的渐近线性性和正态性——这些性质在超高维场景下可能不成立。为克服这些挑战，我们首先将经典的鞅变换扩展到超高维设置下，在温和条件下构造一种基于鞅变换、投影残差标记经验过程的Cramer-von Mises型检验，适用于单位球上的任何投影。鞅变换使这种投影检验渐近无分布，并允许我们仅使用标准的参数估计量收敛速率来推导其极限分布。虽然投影检验在温和条件下对几乎所有单位球上的投影是一致的，但它可能仍会在特定投影上遭受功率损失。因此，我们进一步采用强大的p值组合程序，如Cauchy组合，以在多个投影上汇总p值，从而增强整体鲁棒性。此外，认识到基于经验过程的检验在检测低频信号方面表现优异，而局部平滑检验通常在处理高频替代项时更优，我们提出了一种新的混合检验，使用Cauchy组合将这两种方法结合起来。所得到的混合检验在低频和高频替代项上都具有强大的效力。
英文摘要：
This paper develops a novel methodology for testing the goodness-of-fit of sparse parametric regression models based on projected empirical processes and p-value combination, where the covariate dimension may substantially exceed the sample size. In such ultra-high dimensional settings, traditional empirical process-based tests often fail due to the curse of dimensionality or their reliance on the asymptotic linearity and normality of parameter estimators--properties that may not hold under ultra-high dimensional scenarios. To overcome these challenges, we first extend the classic martingale transformation to ultra-high dimensional settings under mild conditions and construct a Cramer-von Mises type test based on a martingale-transformed, projected residual-marked empirical process for any projection on the unit sphere. The martingale transformation renders this projected test asymptotically distribution-free and enables us to derive its limiting distribution using only standard convergence rates of parameter estimators. While the projected test is consistent for almost all projections on the unit sphere under mild conditions, it may still suffer from power loss for specific projections. Therefore, we further employ powerful p-value combination procedures, such as the Cauchy combination, to aggregate p-values across multiple projections, thereby enhancing overall robustness. Furthermore, recognizing that empirical process-based tests excel at detecting low-frequency signals while local smoothing tests are generally superior for high-frequency alternatives, we propose a novel hybrid test that aggregates both approaches using Cauchy combination. The resulting hybrid test is powerful against both low-frequency and high-frequency alternatives. $\cdots$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
