12,047 research outputs found
Plausibility functions and exact frequentist inference
In the frequentist program, inferential methods with exact control on error
rates are a primary focus. The standard approach, however, is to rely on
asymptotic approximations, which may not be suitable. This paper presents a
general framework for the construction of exact frequentist procedures based on
plausibility functions. It is shown that the plausibility function-based tests
and confidence regions have the desired frequentist properties in finite
samples---no large-sample justification needed. An extension of the proposed
method is also given for problems involving nuisance parameters. Examples
demonstrate that the plausibility function-based method is both exact and
efficient in a wide variety of problems.Comment: 21 pages, 5 figures, 3 table
Minimax and Adaptive Inference in Nonparametric Function Estimation
Since Stein's 1956 seminal paper, shrinkage has played a fundamental role in
both parametric and nonparametric inference. This article discusses minimaxity
and adaptive minimaxity in nonparametric function estimation. Three
interrelated problems, function estimation under global integrated squared
error, estimation under pointwise squared error, and nonparametric confidence
intervals, are considered. Shrinkage is pivotal in the development of both the
minimax theory and the adaptation theory. While the three problems are closely
connected and the minimax theories bear some similarities, the adaptation
theories are strikingly different. For example, in a sharp contrast to adaptive
point estimation, in many common settings there do not exist nonparametric
confidence intervals that adapt to the unknown smoothness of the underlying
function. A concise account of these theories is given. The connections as well
as differences among these problems are discussed and illustrated through
examples.Comment: Published in at http://dx.doi.org/10.1214/11-STS355 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Bayesian astrostatistics: a backward look to the future
This perspective chapter briefly surveys: (1) past growth in the use of
Bayesian methods in astrophysics; (2) current misconceptions about both
frequentist and Bayesian statistical inference that hinder wider adoption of
Bayesian methods by astronomers; and (3) multilevel (hierarchical) Bayesian
modeling as a major future direction for research in Bayesian astrostatistics,
exemplified in part by presentations at the first ISI invited session on
astrostatistics, commemorated in this volume. It closes with an intentionally
provocative recommendation for astronomical survey data reporting, motivated by
the multilevel Bayesian perspective on modeling cosmic populations: that
astronomers cease producing catalogs of estimated fluxes and other source
properties from surveys. Instead, summaries of likelihood functions (or
marginal likelihood functions) for source properties should be reported (not
posterior probability density functions), including nontrivial summaries (not
simply upper limits) for candidate objects that do not pass traditional
detection thresholds.Comment: 27 pp, 4 figures. A lightly revised version of a chapter in
"Astrostatistical Challenges for the New Astronomy" (Joseph M. Hilbe, ed.,
Springer, New York, forthcoming in 2012), the inaugural volume for the
Springer Series in Astrostatistics. Version 2 has minor clarifications and an
additional referenc
A simple recipe for making accurate parametric inference in finite sample
Constructing tests or confidence regions that control over the error rates in
the long-run is probably one of the most important problem in statistics. Yet,
the theoretical justification for most methods in statistics is asymptotic. The
bootstrap for example, despite its simplicity and its widespread usage, is an
asymptotic method. There are in general no claim about the exactness of
inferential procedures in finite sample. In this paper, we propose an
alternative to the parametric bootstrap. We setup general conditions to
demonstrate theoretically that accurate inference can be claimed in finite
sample
A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models
This note uses a simple example to show how moment inequality models used in
the empirical economics literature lead to general minimax relative efficiency
comparisons. The main point is that such models involve inference on a low
dimensional parameter, which leads naturally to a definition of "distance"
that, in full generality, would be arbitrary in minimax testing problems. This
definition of distance is justified by the fact that it leads to a duality
between minimaxity of confidence intervals and tests, which does not hold for
other definitions of distance. Thus, the use of moment inequalities for
inference in a low dimensional parametric model places additional structure on
the testing problem, which leads to stronger conclusions regarding minimax
relative efficiency than would otherwise be possible
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