In logistic regression modeling, Firth's modified estimator is widely used to address the issue of data separation, which results in the nonexistence of the maximum likelihood estimate. Firth's modified estimator can be formulated as a penalized maximum likelihood estimator in which Jeffreys' prior is adopted as the penalty term. Despite its widespread use in practice, the formal verification of the corresponding estimate's existence has not been established. In this study, we establish the existence theorem of Firth's modified estimate in binomial logistic regression models, assuming only the full column rankness of the design matrix. We also discuss other binomial regression models obtained through alternating link functions and prove the existence of similar penalized maximum likelihood estimates for such models.

翻译：在逻辑回归建模中，Firth修正估计被广泛用于解决数据分离问题——该问题会导致最大似然估计不存在。Firth修正估计可表述为一种惩罚最大似然估计，其采用Jeffreys先验作为惩罚项。尽管该方法在实践中应用广泛，但其对应估计量的存在性尚未得到严格验证。本研究仅在设计矩阵满秩的条件下，证明了二项逻辑回归模型中Firth修正估计的存在性定理。同时，我们探讨了通过交替链接函数获得的其他二项回归模型，并证明了此类模型的类似惩罚最大似然估计的存在性。