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修正估计量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修正估计量可表述为一种惩罚最大似然估计量，其中采用Jeffreys先验作为惩罚项。尽管该方法在实际中应用广泛，但其对应估计量存在性的正式验证尚未建立。本研究建立了二项逻辑回归模型中Firth修正估计的存在性定理，仅需假设设计矩阵满列秩。我们还讨论了多项逻辑回归模型。与二项回归情形不同，我们通过实例表明：当参数发散时，Jeffreys先验惩罚项未必发散至负无穷。