Abstract
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 nonidentifiable 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.
Funding Statement
This work was supported by University of Padova grant no. CPDA101912 Large- and small-sample inference under nonstandard conditions (Progetto di Ricerca di Ateneo 2010).
Acknowledgments
Earlier and more extended versions of this article can be found on https://arxiv.org/abs/2206.15178. We thank the reviewers and editors for valuable suggestions that greatly improved the paper, and acknowledge inspiring discussions with Ruggero Bellio, Nancy Reid and Anthony Davison, the last of whom also gave helpful comments on the final version of the manuscript.
Citation
Alessandra R. Brazzale. Valentina Mameli. "Likelihood Asymptotics in Nonregular Settings: A Review with Emphasis on the Likelihood Ratio." Statist. Sci. 39 (2) 322 - 345, May 2024. https://doi.org/10.1214/23-STS910
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