On the Use of Wald's Test in Exponential

Summary Hauck & Donner (1977) showed that Wald's test (the maximum likelihood test statistic) behaves in an aberrant manner when applied to hypotheses about a single parameter in a binomial logit model. In particular they have shown that the test statistic decreases to zero as the parameter estimate moves away from the null value. In the present work the behaviour of Wald's test when applied to hypothesis testing in exponential families is studied. The investigation is mainly restricted to the one-sample problem for one-parameter exponential families. Conditions under which Wald's test is well behaved and conditions under which Wald's test may be misleading are derived. It is shown that the problem occurs in connection with certain parameterizations of discrete probability distributions and also, in the continuous case, if the upper tail of the distribution function is approximately proportional to t-e-e' for some positive 0. Finally, the use of Wald's test in the analysis of generalized linear models is discussed.