by Gaffke, N., Heiligers, B., Offinger, R..

**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.