### 2001-16

#### On the Asymptotic Null-Distribution of the Wald Statistic at Singular Parameter Points

Series: 2001-16, Preprints

MSC:
62F05 Asymptotic properties of tests
62J05 Linear regression

Abstract:
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 a 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.

Keywords:
Nonlinear hypothesis, asymptotically normal estimator, Wald test, unconfounded regression.

This paper was published in:
Statistics & Decisions 20 (2002), 379-398.