Recently I was very embarrassed when I gave an off the cuff answer about minimum variance unbiased estimates for parameters of a uniform distribution that was completely wrong. Fortunately I was immediately corrected by cardinal and Henry with Henry providing the correct answers for the OP.
This got me thinking though. I learned the theory of best unbiased estimators in my graduate math stat class at Stanford some 37 years ago. I have recollections of the Rao-Blackwell theorem, the Cramer - Rao lower bound and the Lehmann-Scheffe Theorem. But as an applied statistician I don't think very much about UMVUEs in my daily life whereas maximum likelihood estimation comes up a lot.
Why is that? Do we overemphasize the UMVUE theory too much in graduate school? I think so. First of all unbiasedness is not a crucial property. Many perfectly good MLEs are biased. Stein shrinkage estimators are biased but dominate the unbiased MLE in terms of mean square error loss. It is a very beautiful theory (UMVUE estimation), but very incomplete and I think not very useful. What do others think?