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Testing the goodness-of-fit of a model with its defining functionalconstraints in the parameters could date back to Spearman (1927), who analyzedthe famous "tetrad" polynomial in the covariance matrix of the observedvariables in a single-factor model. Despite its long history, the Wald testtypically employed to operationalize this approach could produce veryinaccurate test sizes in many situations, even when the regular conditions forthe 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 nullhypothesis is true but "near-singular", the standardized Wald test exhibitsslow weak convergence, owing to the sophisticated dependency structure inherentto the underlying U-statistic that ultimately drives its limiting distribution;this can also be rigorously explained by a key ratio of moments encoded in theBerry-Esseen bound quantifying the normal approximation error involved. As analternative, we advocate the use of an incomplete U-statistic to mildly tonedown the dependence thereof and render the speed of convergence agnostic to thesingularity status of the hypothesis. In parallel, we develop a Berry-Esseenbound that is mathematically descriptive of the singularity-agnostic nature ofour standardized incomplete U-statistic, using some of the finestexponential-type inequalities in the literature.

Singularity-agnostic incomplete U-statistics for testing polynomial constraints in Gaussian covariance matrices