A reminder of the fallibility of the Wald statistic: Likelihood explanation

The Wald statistic is one of the most commonly used tools in applied statistics, so it is sobering to read Fears, Benichou, and Gall's recent reminder of its fallibility. What makes their example particularly relevant is the fact that the problem is manifest in a simple normal random effects model on a balanced dataset for a seemingly harmless parameter, while, in practice, one tends to rely on the Wald test in complicated or nonnormal models where there are no exact tests to serve as a gold standard. This article explains the failure of the Wald test via the profile likelihood functions, which graphically look decidedly nonnormal. The methodology used to derive the profile likelihood and the discussion of this example could be instructive for a senior or beginning graduate class in theory of statistics.