用于检验高斯协方差矩阵中多项式约束的奇异性无关的不完全U-统计量
Singularity-agnostic incomplete U-statistics for testing polynomial constraints in Gaussian covariance matrices
AI总结：
针对高斯协方差矩阵多项式约束检验中Wald检验在近奇异原假设下不准确的问题，提出使用不完全U-统计量来缓解依赖结构，使收敛速度与假设的奇异性无关，并开发了相应的Berry-Esseen界限。
摘要：
检验模型与其参数中定义的函数约束的拟合优度可以追溯到Spearman (1927)，他分析了单因子模型中观测变量协方差矩阵里著名的“tetrad”多项式。尽管历史悠久，但通常用于实现该方法的Wald检验在许多情况下可能会产生非常不准确的检验水平，即使满足了经典正态渐近的常规条件且有非常大的样本可用。专注于检验高斯协方差矩阵中的多项式约束，我们对这一令人困惑的现象有了新的理解：当原假设为真但“近奇异”时，标准化的Wald检验表现出缓慢的弱收敛，这是由于最终驱动其极限分布的底层U-统计量固有的复杂依赖结构所致；这也可以通过量化所涉及正态近似误差的Berry-Esseen界限中编码的关键矩比来严格解释。作为替代，我们主张使用不完全U-统计量来适度缓和其依赖性，并使收敛速度与假设的奇异性状态无关。同时，我们利用文献中一些最精细的指数型不等式，开发了一个Berry-Esseen界限，在数学上描述了我们的标准化不完全U-统计量的奇异性无关性质。
英文摘要：
Testing the goodness-of-fit of a model with its defining functional constraints in the parameters could date back to Spearman (1927), who analyzed the famous "tetrad" polynomial in the covariance matrix of the observed variables in a single-factor model. Despite its long history, the Wald test typically employed to operationalize this approach could produce very inaccurate test sizes in many situations, even when the regular conditions for the classical normal asymptotics are met and a very large sample is available. Focusing on testing a polynomial constraint in a Gaussian covariance matrix, we obtained a new understanding of this baffling phenomenon: When the null hypothesis is true but "near-singular", the standardized Wald test exhibits slow weak convergence, owing to the sophisticated dependency structure inherent to the underlying U-statistic that ultimately drives its limiting distribution; this can also be rigorously explained by a key ratio of moments encoded in the Berry-Esseen bound quantifying the normal approximation error involved. As an alternative, we advocate the use of an incomplete U-statistic to mildly tone down the dependence thereof and render the speed of convergence agnostic to the singularity status of the hypothesis. In parallel, we develop a Berry-Esseen bound that is mathematically descriptive of the singularity-agnostic nature of our standardized incomplete U-statistic, using some of the finest exponential-type inequalities in the literature.
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。
