Statistics, Conditional distributions, UMVUE, Rao-Blackwell

[libragirl79]

Hi,

I have a general concept question.
I am working with finding complete sufficient statistics of distributions. Sometimes I need to condition some function of a parameter on a sufficient statistic, using basically Rao-Blackwell, but my trouble is in finding the conditional distributions so I can get the mean.

For example, if Xbar denotes the mean of some random sample X1,X2,...Xn from a gamma distribution where alpha is known and beta is the parameter I am trying to estimate, I know that Xbar is the sufficient statistic and choosing X1 as my preliminary estimator, I would condition it on Xbar and get the mean, so I am trying to find:
E[X1|Xbar]. My question is how do I find the distribution of X1|Xbar? I know that conditional distribution is joint of (X1 and Xbar) divided by pdf of Xbar, but I am not sure how to actually go about this...

Another example would be if I have a pdf: e^-(x-θ) and want to find the best unbiased estimator for θ^r. So, given that I know that smallest order statistic X(1) is my complete sufficient statistic, I am basically looking for E[some function u(x)|X(1)]=θ^r and to find that function, need to get the conditional distribution of u(x) given X(1).

I hope this makes sense of what I am trying to do...

Thanks very much for any help!

 

[abeliando]

Hm. Don't you want an unbiased estimator to start with? Is X₁ unbiased for β?

 

[libragirl79]

No, X1 wouldn't be an unbiased estimator of Beta, since the mean of X1 is alpha*beta given that X1 has gamma dist...

 

[abeliando]

Yeah, it seems a bit difficult. If you're looking for the UMVUE, you don't have to though. Can't you use Lehmann-Scheffe here, in the case of the Gamma?

 

[libragirl79]

yeah, I believe if alpha is given, then the MLE for Gamma would be Xbar/alpha. I am just not sure how to get these conditional distributions when doing Rao-Blackwellizing...