Likelihood Asymptotics in Nonregular Settings. A Review with Emphasis on the Likelihood Ratio

We review the most common situations where one or some of the regularity conditions which underlie likelihood based parametric inference fail. Three main classes of problems will be treated: boundary problems, indeterminate parameters and singular information matrices, and change-point problems. The review is focused on the large- and small-sample properties of the likelihood ratio, though other approaches to hypothesis testing and connections to estimation will be mentioned in passing.