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.

翻译：基于模型参数的功能性约束进行模型拟合优度检验可追溯至Spearman(1927)对单因子模型观测变量协方差矩阵中著名"四元组"多项式的分析。尽管该检验方法历史悠久，但通常用于实现这一方法的Wald检验在许多情况下会产生极不准确的检验水平——即便满足经典正态渐近性的正则条件且样本容量极大时也是如此。本文聚焦于高斯协方差矩阵中多项式约束的检验，对这一令人困惑的现象获得了新的认识：当原假设为真但"近奇异"时，标准化Wald检验呈现缓慢弱收敛性，其根源在于决定极限分布的底层U统计量固有的复杂依赖结构；这种收敛缓慢现象也可通过控制正态逼近误差的Berry-Esseen界中编码的关键矩比率得到严格解释。作为替代方案，我们建议采用非完全U统计量适度削弱其依赖性，使收敛速度与原假设的奇异状态无关。与此同时，我们利用文献中最精细的指数型不等式，发展了一个在数学上精确描述标准化非完全U统计量奇异性无关性质的Berry-Esseen界。