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Re: st: types of standard error
On May 23, 2006, at 9:03 AM, [email protected] wrote:
For example, in glm, we have the option of using:
OIM, EIM, OPG, HAC, jacknife, one-stepped jacknife, unbiased
sandwich...
If you really want to learn more about the differences between these
estimators, that's great; be forewarned, however, you're not going to
get very far without delving into some of the math. The distinction
between the expected information (EIM) and observed information (OIM)
is an important one, and comes out of basic likelihood theory (any
basic book on statistical theory will discuss this). In some cases
(e.g., generalized linear models with canonical link), they are
equivalent. A classic paper comparing the two is "Assessing the
accuracy of the maximum likelihood estimator: Observed versus
expected Fisher information" (Efron and Hinkley, Biometrika (65)
1978). This paper argues that the observed information is in general
more appropriate, which (I presume) is why -glm- produces this by
default.
The robust (sandwich) estimator is an attempt to relax the model
assumptions on which the standard calculation is based. For a simple
introduction (with examples), see "Model Robust Confidence Intervals
Using Maximum Likelihood Estimators" (Royall, International
Statistical Review (54) 1986). Also, as I said before, see [U] 20.14
and the references cited therein.
Efron and Tibshirani's book on the bootstrap (the one I suggested
earlier) actually treats both the bootstrap and the jackknife in
detail, and discusses the relationship between them and several other
estimates of variance (e.g., expected and observed information,
sandwich, etc.). In this respect, it is one of the better (and more
accessible) treatments of the differences among the various estimators.
Finally, if you're primarily interested in this issue WRT -glm-, I
believe Hardin and Hilbe's book (http://www.stata.com/bookstore/
glmext.html) has a good discussion of the differences between the
various estimators available.
Keep in mind, more choice (and the ease with which the different
estimates may be obtained and compared) is a very good thing. You
can never go wrong by trying the different estimators to verify that
they give similar results (which they often do). In cases where the
results are substantially different, however, you need to be careful
picking one over the others. And as always, they justifications for
many of these estimators rely on asymptotic arguments, and therefore
you should always be careful when applying them to data from small
samples.
-- Phil
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