  Motivated by the problem of testing tetrad constraints in factor analysis, we
study the large-sample distribution of Wald statistics at parameter points at
which the gradient of the tested constraint vanishes. When based on an
asymptotically normal estimator, the Wald statistic converges to a rational
function of a normal random vector. The rational function is determined by a
homogeneous polynomial and a covariance matrix. For quadratic forms and
bivariate monomials of arbitrary degree, we show unexpected relationships to
chi-square distributions that explain conservative behavior of certain Wald
tests. For general monomials, we offer a conjecture according to which the
reciprocal of a certain quadratic form in the reciprocals of dependent normal
random variables is chi-square distributed.
