Selfinformative limits of Bayes estimates and generalized maximum likelihood

Abstract

A definition of selfinformative Bayes carriers or limits is given as a description of an approach to non-informative Bayes estimation in non- and semiparametric models. It takes the posterior w.r.t. a prior as a new prior and repeats this procedure again and again. A main objective of this article is to clarify the relation between selfinformative carriers or limits and maximum likelihood estimates (MLEs). For a model with dominated probability distributions, we state sufficient conditions under which the set of MLEs is a selfinformative carrier or in the case of a unique MLE its selfinformative limit property. Mixture models are covered. The result on carriers is extended to more general models without dominating measure. Selfinformative limits, in the case of estimation, of hazard functions based in censored observations and in the case of normal linear models with possibly non-identifiable parameters are shown to be identical to the generalized MLEs in the sense of Gill [Gill, R.D., 1989, Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. Scandinanian Journal of Statistics, 16(2), 97–128.] and Kiefer and Wolfowitz [Kiefer, J. and Wolfowitz, J., 1956, Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics, 27, 887–906.]. Selfinformative limits are given for semiparametric linear models. For a location model, they are identical to generalized MLEs, while this is not true in general.