On Newton's Method for Entire Functions

The Newton map N_f of an entire function f turns the roots of f into attracting fixed points. Let U be the immediate attracting basin for such a fixed point of N_f. We study the behavior of N_f in a component V of C\U. If V can be surrounded by an invariant curve within U and satisfies the condition that each point in the extended plane has at most finitely many preimages in V, we show that V contains another immediate basin of N_f or a virtual immediate basin. A virtual immediate basin is an unbounded invariant Fatou component in which the dynamics converges to infty through an absorbing set.Comment: 19 pages, 4 figures. Changes in Version 2: Sharpened the result in Section

Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models

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