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

This paper reviews the most common situations in which the regularity conditions that underlie classical likelihood-based parametric inference fail, focusing on the large-sample properties of the likelihood ratio statistic. 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. We emphasise analytical solutions, consider software implementations where available, and summarise how the key results are derived.