The Geometry of Existence and Uniqueness of Maximum Likelihood Estimation in Categorical Response Models
Abstract
Nonexistence of the maximum likelihood estimate (MLE) under separation is treated as a solved problem for binary logistic regression and as a scattered collection of model-specific results everywhere else. We show that it is one phenomenon with one criterion. In a latent polyhedral categorical response model, every observed outcome corresponds to a polyhedral event in latent variables whose faces shift linearly with the parameter. Random-utility choice, cumulative-link, ranking, multivariate binary and ordinal, sequential, adjacent-category logit, and fixed-score stereotype models belong to this family. The likelihood sees each observed factor only through the columns of its threshold map, the structure vectors. A finite MLE exists if and only if the pooled structure vector set has overlap. Sufficiency requires only continuity, necessity requires only strictly threshold-increasing probabilities, and neither likelihood concavity, exchangeability, nor full design rank is needed. Positive strictly log-concave latent densities and threshold-identifiable polyhedra then yield uniqueness on the estimable span. A single linear program determines whether overlap holds, and convex cone geometry measures the dimension of separation. An application using cumulative-link models for willingness to share health data identifies observations and model terms associated with nonexistence and illustrates how the diagnostics inform model revision and sensitivity analysis.
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BibTeX
@misc{sablica2026geometry,
title = {{The Geometry of Existence and Uniqueness of Maximum Likelihood Estimation in Categorical Response Models}},
author = {Lukas Sablica and Kurt Hornik and Thomas Rusch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08100},
eprint = {2610.08100},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08100},
}
RIS
TY - UNPB
TI - The Geometry of Existence and Uniqueness of Maximum Likelihood Estimation in Categorical Response Models
AU - Lukas Sablica
AU - Kurt Hornik
AU - Thomas Rusch
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.08100
UR - https://arxiv.org/abs/2610.08100
AB - Nonexistence of the maximum likelihood estimate (MLE) under separation is treated as a solved problem for binary logistic regression and as a scattered collection of model-specific results everywhere else. We show that it is one phenomenon with one criterion. In a latent polyhedral categorical response model, every observed outcome corresponds to a polyhedral event in latent variables whose faces shift linearly with the parameter. Random-utility choice, cumulative-link, ranking, multivariate binary and ordinal, sequential, adjacent-category logit, and fixed-score stereotype models belong to this family. The likelihood sees each observed factor only through the columns of its threshold map, the structure vectors. A finite MLE exists if and only if the pooled structure vector set has overlap. Sufficiency requires only continuity, necessity requires only strictly threshold-increasing probabilities, and neither likelihood concavity, exchangeability, nor full design rank is needed. Positive strictly log-concave latent densities and threshold-identifiable polyhedra then yield uniqueness on the estimable span. A single linear program determines whether overlap holds, and convex cone geometry measures the dimension of separation. An application using cumulative-link models for willingness to share health data identifies observations and model terms associated with nonexistence and illustrates how the diagnostics inform model revision and sensitivity analysis.
ER -