13,114 research outputs found
Inferential models: A framework for prior-free posterior probabilistic inference
Posterior probabilistic statistical inference without priors is an important
but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of
belief functions, and Bayesian inference with default priors are attempts to
achieve this goal but, to date, none has given a completely satisfactory
picture. This paper presents a new framework for probabilistic inference, based
on inferential models (IMs), which not only provides data-dependent
probabilistic measures of uncertainty about the unknown parameter, but does so
with an automatic long-run frequency calibration property. The key to this new
approach is the identification of an unobservable auxiliary variable associated
with observable data and unknown parameter, and the prediction of this
auxiliary variable with a random set before conditioning on data. Here we
present a three-step IM construction, and prove a frequency-calibration
property of the IM's belief function under mild conditions. A corresponding
optimality theory is developed, which helps to resolve the non-uniqueness
issue. Several examples are presented to illustrate this new approach.Comment: 29 pages with 3 figures. Main text is the same as the published
version. Appendix B is an addition, not in the published version, that
contains some corrections and extensions of two of the main theorem
Bias in parametric estimation: reduction and useful side-effects
The bias of an estimator is defined as the difference of its expected value
from the parameter to be estimated, where the expectation is with respect to
the model. Loosely speaking, small bias reflects the desire that if an
experiment is repeated indefinitely then the average of all the resultant
estimates will be close to the parameter value that is estimated. The current
paper is a review of the still-expanding repository of methods that have been
developed to reduce bias in the estimation of parametric models. The review
provides a unifying framework where all those methods are seen as attempts to
approximate the solution of a simple estimating equation. Of particular focus
is the maximum likelihood estimator, which despite being asymptotically
unbiased under the usual regularity conditions, has finite-sample bias that can
result in significant loss of performance of standard inferential procedures.
An informal comparison of the methods is made revealing some useful practical
side-effects in the estimation of popular models in practice including: i)
shrinkage of the estimators in binomial and multinomial regression models that
guarantees finiteness even in cases of data separation where the maximum
likelihood estimator is infinite, and ii) inferential benefits for models that
require the estimation of dispersion or precision parameters
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