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Shrinkage Confidence Procedures
The possibility of improving on the usual multivariate normal confidence was
first discussed in Stein (1962). Using the ideas of shrinkage, through Bayesian
and empirical Bayesian arguments, domination results, both analytic and
numerical, have been obtained. Here we trace some of the developments in
confidence set estimation.Comment: Published in at http://dx.doi.org/10.1214/10-STS319 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A default prior for regression coefficients
When the sample size is not too small, M-estimators of regression
coefficients are approximately normal and unbiased. This leads to the familiar
frequentist inference in terms of normality-based confidence intervals and
p-values. From a Bayesian perspective, use of the (improper) uniform prior
yields matching results in the sense that posterior quantiles agree with
one-sided confidence bounds. For this, and various other reasons, the uniform
prior is often considered objective or non-informative. In spite of this, we
argue that the uniform prior is not suitable as a default prior for inference
about a regression coefficient in the context of the bio-medical and social
sciences. We propose that a more suitable default choice is the normal
distribution with mean zero and standard deviation equal to the standard error
of the M-estimator. We base this recommendation on two arguments. First, we
show that this prior is non-informative for inference about the sign of the
regression coefficient. Secondly, we show that this prior agrees well with a
meta-analysis of 50 articles from the MEDLINE database
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