AI 中文总结
本文揭示分类响应模型中MLE存在性与唯一性可由统一几何判据刻画，通过结构向量重叠条件与线性规划判定，并应用于健康数据共享模型诊断。
AI 中文摘要
在分离（separation）条件下最大似然估计（MLE）的不存在性，对于二元逻辑回归被视为已解决的问题，而在其他场景中则被视为一系列针对特定模型的结果。我们证明这是一个具有单一判据的统一现象。在潜在多面体分类响应模型中，每个观测结果对应于潜在变量中的一个多面体事件，其面随参数线性移动。随机效用选择、累积链接、排序、多元二元与有序、序贯、相邻类别对数几率以及固定得分刻板模型均属于该族。似然函数仅通过每个观测因子的阈值映射的列（即结构向量）来感知该因子。有限MLE存在当且仅当合并的结构向量集合具有重叠。充分性仅需连续性，必要性仅需严格阈值递增概率，且不需要似然凹性、可交换性或完全设计秩。正的严格对数凹潜在密度和阈值可识别的多面体随后在可估跨度上产生唯一性。单个线性规划即可判定重叠是否成立，凸锥几何度量分离的维度。一项使用累积链接模型分析健康数据共享意愿的应用，识别了与不存在性相关的观测和模型项，并展示了诊断如何为模型修正和敏感性分析提供信息。
英文摘要
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.
