Message from Professor Bozenna Pasik-Duncan: 4/8/13
Dear Students,
Let me make one more comment about Rao-Blackwell theorem.
The importance of this theorem is that it shows that for every other unbiased estimator of parameter, we can always find an unbiased estimator based on the sufficient statistic which has a smaller variance than the first unbiased estimator. One based on sufficient statistic is better than the first one. We might begin our search for an unbiased estimator with the smallest variance by considering only those unbiased estimators based on the sufficient statistics. It is also known that if pdf is of the exponential form then if an unbiased estimator exists there is only one function of the sufficient statistic that is unbiased. See the definition of MVUE, p.375.
Proof of R-B theorem can be found at:
http://www.cc.gatech.edu/~lebanon/notes/RaoBlackwell.pdf
or at:
http://www.stats.ox.ac.uk/~steffen/teaching/bs2siMT04/si3c.pdf
and useful remarks about the theorem can be found at:
http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_rao_blackwell.htm