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Re: Constrained estimation
Posted:
Mar 14, 2012 4:20 AM


"Paul" <paulvonhippel@yahoo.com> wrote in message news:945c6d4595954c3aa85830e70564aae8@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/(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!
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



