Wald tests of singular hypotheses

TL;DR

This paper investigates the asymptotic distribution of Wald tests at singular points, revealing new relationships to chi-square distributions and proposing conjectures for general cases.

Contribution

It characterizes the large-sample behavior of Wald statistics at singular hypotheses, including explicit forms and conjectures for their distributions.

Findings

Wald statistic converges to a rational function of a normal vector.

Unexpected links between quadratic forms and chi-square distributions.

Conjecture on the distribution of reciprocals of quadratic forms in normal variables.

Abstract

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.