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Topic: Constrained estimation
Replies: 9   Last Post: Mar 14, 2012 4:38 AM

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Ray Koopman

Posts: 3,382
Registered: 12/7/04
Re: Constrained estimation
Posted: Mar 14, 2012 3:47 AM
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On Mar 13, 9:50 pm, Paul <paulvonhip...@yahoo.com> wrote:
> Thanks for all your suggestions! Several of you asked for assumptions
> or simplifications, so let me try posing a slightly simpler problem
> with clear assumptions:
>
> Let B = b U/(D-1) where b is a constant and U is a central chi-square
> variable with D-1 degrees of freedom. Then E(B) = b.
>
> I happen to know that b < w, where w is a constant; in fact, in most
> settings b is likely to be substantially less than w. However, given a
> sample of n observations on B, it is quite possible for the sample
> mean to exceed w. So the sample mean is not a good estimator; nor is
> the minimum of the sample mean and w.
>
> What are some good ways to estimate b? A good estimate will never be
> equal to or greater than w, and will rarely be close to w. This
> probably means that the estimate will be negatively biased but less
> variable than the sample mean, and hopefully with a lower MSE than the
> sample mean.
>
> I should say that I already have a solution: I use the posterior mean
> of B where the posterior has been truncated on the right at w.
> However, the expression for the posterior mean is a bit nasty, and I
> don't know if it comes close to having minimal MSE.
> I wonder if there are other approaches that give a simpler result or
> one with smaller MSE.
>
> Many thanks for further suggestions. I appreciate your willingness to
> brainstorm!


No answers, just some more questions.

Do you know D, or are you estimating it, or are you integrating it
out?

Why do you say that Min[sample mean, w] is not a good estimator?

You're using Bayes to constrain b < w, but you haven't said what
your prior is in (0,w). I take your comment "in most settings b
is likely to be substantially less than w", as suggesting that
the prior ought to -> 0 as b -> w. Would Log[w/b]/w work for you?

Whatever you do ought to be scale-equivariant; that is, if you
multiply w and all your data by an arbitrary constant c, the
corresponding estimate of b should also be multiplied by c.



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