  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.
