Summary: Consider a null hypothesis \(H_0:R(\theta)=0\) about the parameter vector \(\theta\) in a statistical model, where \(R\) is a given smooth multivariate function. The standard asymptotics for the null-distribution of the Wald statistic (as well as that of other test statistics, e.g., the maximum likelihood ratio statistic), applies when the Jacobian \(J(\theta)\) of \(R\) has full row rank. There are, however, examples of important models and hypotheses which do not meet this assumption, i.e., there are parameter points within the null hypothesis at which the Jacobian is rank deficient, and the standard chi-square asymptotics breaks down at such ‘singular points’. This is not caused by singularity of the information matrix, but is implied by the geometric structure of the null hypothesis.
We derive the asymptotic distributions of the Wald statistics at singular points of the null hypothesis, under a second order regularity condition. The results are illustrated for the null hypothesis of unconfoundedness of a regression \(Y\) on \(X\) w.r.t. a potential confounder \(W\), where \(Y\), \(X\), and \(W\) are dichotomous random variables. Surprisingly, the Wald test using the standard critical \(\chi^2\)-value remains asymptotically conservative if a particular (though fairly natural) formulation of the null hypothesis is used, while for another representation of the null hypothesis this is not true. Further examples considered are the hypothesis of collapsibility of a \(2\times 2\times 2\) table and that of singularity of a matrix-valued parameter.