Understanding Wald's Test for Exponential Families

Anomalous behavior of Ward's test for exponential families is explained on the basis of the use of parameter estimates rather than null values in the variance part of the Wald expression.Illustration of the difficulty is made through use of single-parameter situations, albeit multiparameter situations were first cited to bring out the difficulty.Situations had been claimed in which statistical significance disappeared as data became more inconsistent with the null hypothesis.It is shown here that with use of the Wald test, according to the parameterization employed, the same data can be both consistent with all possible null values for the parameter and inconsistent with all possible values.