
Re: Constrained estimation
Posted:
Mar 14, 2012 3:47 AM


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/(D1) where b is a constant and U is a central chisquare > variable with D1 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 scaleequivariant; 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.

