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 multinomial logistic regression models. Unlike the binomial regression case, we show through an example that the Jeffreys-prior penalty term does not necessarily diverge to negative infinity as the parameter diverges.

翻译：在逻辑回归建模中，Firth 修改的估计器被广泛用于解决数据分离问题，该问题导致最大似然估计不存在。Firth 修改的估计器可以被公式化为一个惩罚的最大似然估计量，其中采用 Jeffreys 先验作为惩罚项。尽管在实践中广泛使用，但相应估计存在性的正式验证尚未确立。在本研究中，我们在假定设计矩阵具有完整列秩的情况下，建立了二项逻辑回归模型中 Firth 修改的估计存在定理。我们还讨论了多项逻辑回归模型。与二项回归情况不同，我们通过一个例子展示，Jeffreys 先验惩罚项不一定会随着参数的发散而发散到负无穷。