Stat 8112 Lecture Notes The Wilks, Wald, and Rao Tests Charles J. Geyer September 26, 2020
1 Introduction
One of the most familiar of results about maximum likelihood is that the likelihood ratio test statistic has an asymptotic chi-square distribution. Those who like eponyms call this the Wilks theorem and the hypothesis test 1 using this test statistic the Wilks test. Let θˆn be the MLE for a model and ∗ θn the MLE for a smooth submodel. The Wilks test statistic is ˆ ∗ Tn = 2 ln(θn) − ln(θn) (1) and the Wilks theorem says
w Tn −→ ChiSq(p − d), where p is the dimension of the parameter space of the model and d is the dimension of the parameter space of the submodel. ˆ ∗ It is sometimes easy to calculate one of θn and θn and difficult or impos- sible to calculate the other. This motivates two other procedures that are asymptotically equivalent to the Wilks test. The Rao test statistic is