  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.
