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PreprintCC BY 4.0arXiv

Asymptotic Theory for Restricted Maximum Likelihood Estimators in Generalized Linear Mixed Models

  • Zhanzhongyu Gao
  • Huadong Mo
  • Ziyang Lyu
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Abstract

Developing an asymptotic theory for restricted maximum likelihood (REML) estimators in generalized linear mixed models (GLMMs) has remained an open problem for decades. In this paper, we establish the asymptotic distributions of the REML and maximum likelihood (ML) estimators for GLMMs when both the number of clusters and the cluster sizes tend to infinity. Under very mild conditions, requiring only finite-moment assumptions on the random effects rather than normality and imposing no restriction on the relative rates at which the number of clusters and the cluster sizes diverge, both estimators are proved to be asymptotically normal with an explicit block-diagonal covariance structure, and the bias-correction property of REML is established. Simulation studies support the asymptotic results, and a genetic data analysis illustrates the methodology.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026asymptotic,
  title = {{Asymptotic Theory for Restricted Maximum Likelihood Estimators in Generalized Linear Mixed Models}},
  author = {Zhanzhongyu Gao and Huadong Mo and Ziyang Lyu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13862},
  eprint = {2609.13862},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13862},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotic Theory for Restricted Maximum Likelihood Estimators in Generalized Linear Mixed Models
AU  - Zhanzhongyu Gao
AU  - Huadong Mo
AU  - Ziyang Lyu
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13862
UR  - https://arxiv.org/abs/2609.13862
AB  - Developing an asymptotic theory for restricted maximum likelihood (REML) estimators in generalized linear mixed models (GLMMs) has remained an open problem for decades. In this paper, we establish the asymptotic distributions of the REML and maximum likelihood (ML) estimators for GLMMs when both the number of clusters and the cluster sizes tend to infinity. Under very mild conditions, requiring only finite-moment assumptions on the random effects rather than normality and imposing no restriction on the relative rates at which the number of clusters and the cluster sizes diverge, both estimators are proved to be asymptotically normal with an explicit block-diagonal covariance structure, and the bias-correction property of REML is established. Simulation studies support the asymptotic results, and a genetic data analysis illustrates the methodology.
ER  -