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Re: Reducing bias of a Bayesian point estimator
Posted:
Oct 18, 2012 6:19 PM
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On Thursday, October 18, 2012 8:13:11 AM UTC-5, David Jones wrote:
> "Paul" wrote in message
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> news:a9bda0dc-b6c3-4105-a9c6-d1af75e58d2e@googlegroups.com...
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> > > > i.e.,
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> >
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> >
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> > Many of the usual frequentist techniques for bias adjustment would lead
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> > to estimates not obeying your bound. If you were not insisting on not
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> > allowing a value of gamma exactly equal to one then the obvious thing to
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> > start from
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> > would be a simple truncation at 1. You might consider truncating at a
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> > value slightly smaller than one if an exact-1 causes problems. Otherwise
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> > you
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> > might be able to base something on he following....
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> >
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> > (a) let T be the non-truncated version of the estimated for gamma;
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> >
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> > (b) the transformed estimate S=T/(1+T) should have good properties if the
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> > true value of gamma is small, and is forced to lie in the required
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> > interval;
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> >
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> > (c) consider the extended set of estimates S=T/(1+T^a)^(1/a), which are
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> > again forced to lie in 0 to 1, and choose a for good properties
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> >
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> > Other families of transformed estimates might be considered.
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> This is an intriguing possibility except that what I need is an estimator
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> that performs well if the true value of gamma is large (close to 1) not
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> small.
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> --------------------------------------------------------------------------------
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> The basis here is that such an approach starts from an estimating function
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> that has good properties both for small gamma, and for large (since the
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> estimators are constrained to be close to 1 if the raw sample estimate is
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> large). It is not clear how much can be achieved analytically, but a
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> minimum might be to use an estimator S=bT/(1+bT), and choose b to that the
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> bias is correct to a low order in a power series expansion in terms of
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> gamma. This would notionally spread the good property of the estimator near
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> zero, away from zero to some unknown extent ... you might hope to get an
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> explicit formula for a value of b in terms of your sample size N.
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> Alternatively, you might allow a clipped estimator S= min(1, bT) and see
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> what arises ... you might get some formulae in terms of incomplete gamma
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> functions, valid for any gamma, and then see if you are happy with the
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> result for gamma=1. You were not clear whether a value for the estimator
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> exactly equal to one causes problems (for your confidence intervals)...
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> which is a different question from whether gamma=1 is valid within the
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> model.
Thank you. I haven't seen estimators of the form S=bT/(1+bT) before. Is there a name for that class of estimators? Perhaps I should read more about it.
An estimate of 1 for gamma is unfortunately not usable, though I have experimented with estimators of the form S= min(1-e, bT) where e is small.
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