Ordinarily, Wald's likelihood ratio statistic is essentially the same as Wilks's statistic, which in one-parameter problems, is the squared ratio of the estimate to its standard error. But there are exceptional cases where a substantial discrepancy may occur, and variance-components models provide good examples. In order to understand the source of the discrepancy, simulate data with simple structure as follows:

The null hypothesis is that \(\mu \propto \mathbf{1}\) is constant, and the alternative is that \(\mu=X \beta\) for some \(\beta\) with non-negative components. Test this hypothesis using Wilks's likelihood ratio statistic, and also using the Wald statistic. Recall the exponential assumption, which implies that the dispersion parameter is one.

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set.seed (3142); n < rx2 < 1/x^2; beta < 1000; x < 1:n c (1, 20); X