  Penalization of the likelihood by Jeffreys' invariant prior, or by a positive
power thereof, is shown to produce finite-valued maximum penalized likelihood
estimates in a broad class of binomial generalized linear models. The class of
models includes logistic regression, where the Jeffreys-prior penalty is known
additionally to reduce the asymptotic bias of the maximum likelihood estimator;
and also models with other commonly used link functions such as probit and
log-log. Shrinkage towards equiprobability across observations, relative to the
maximum likelihood estimator, is established theoretically and is studied
through illustrative examples. Some implications of finiteness and shrinkage
for inference are discussed, particularly when inference is based on Wald-type
procedures. A widely applicable procedure is developed for computation of
maximum penalized likelihood estimates, by using repeated maximum likelihood
fits with iteratively adjusted binomial responses and totals. These theoretical
results and methods underpin the increasingly widespread use of reduced-bias
and similarly penalized binomial regression models in many applied fields.
