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

This paper reviews the most common situations where one or more regularity conditions which underlie classical likelihood-based parametric inference fail. We identify three main classes of problems: boundary problems, indeterminate parameter problems--which include non-identifiable parameters and singular information matrices--and change-point problems. The review focuses on the large-sample properties of the likelihood ratio statistic, though other approaches to hypothesis testing and connections to estimation may be mentioned in passing. We emphasize analytical solutions and acknowledge software implementations where available. Some summary insight about the possible tools to derivate the key results is given.