Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.stat.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
David Jones

Posts: 324
Registered: 2/28/07
Re: Constrained estimation
Posted: Mar 14, 2012 4:20 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

"Paul" <paulvonhippel@yahoo.com> wrote in message
news:945c6d45-9595-4c3a-a858-30e70564aae8@w5g2000yqi.googlegroups.com...
> 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




Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.