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

 Messages: [ Previous | Next ]
 David Jones Posts: 324 Registered: 2/28/07
Re: Constrained estimation
Posted: Mar 14, 2012 4:20 AM

"Paul" <paulvonhippel@yahoo.com> wrote in message
> 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!

For the unconstrained case, there are known results for a multiplying factor
to apply to B to give the minimum mean square error estimate under your
assumptions, so you could hope to find something that will reduce to this
result. But you seem to be wanting to impose a lot of information that the
"true" value b will be a lot less than the bound w. Therefore it seems that
you would need to find an at least moderately informative prior that will
represent what you "know". Possibly a beta distribution might suit, and
there just might be a special case that would allow an analytical result to
be derived.

David Jones

Date Subject Author
3/12/12 paulvonhippel at yahoo
3/12/12 Ray Koopman
3/12/12 paulvonhippel at yahoo
3/12/12 Ray Koopman
3/12/12 Herman Rubin
3/12/12 David Jones
3/14/12 paulvonhippel at yahoo
3/14/12 Ray Koopman
3/14/12 David Jones
3/14/12 Ray Koopman