I am confused about this. Are they the same? There are some problems in which I am asked to find the Uniformly Minimum Variance Unbiased Estimator (UMVUE) first and then check if it achieves the CRLB. Yet from what I've read, I understand that an unbiased estimator that achieves CRLB is UMVUE. Thanks.
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$\begingroup$ stats.stackexchange.com/questions/436384/… $\endgroup$– StubbornAtomCommented Jan 13, 2020 at 16:29
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They are not the same. Achieving the CR lower bound is a sufficient condition for an unbiased estimator to be UMVUE. However, it is not necessary.
For example, Example 3.10 in this link gives an estimator that is UMVUE (by the Lehmann-Scheffe theorem) but does not attain the CR Lower bound.
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$\begingroup$ Is there a general procedure for finding UMVUEs without the use of CRLB? $\endgroup$ Commented Jul 6, 2017 at 1:02
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1$\begingroup$ @user164144 Yes, if the unbiased estimator is a function of a complete sufficient statistic, then it is the UMVUE. This is the Lehmann-Scheffe theorem. In particular, let $T$ be a complete sufficient statistic and $S$ an unbiased estimator. Then $E[S | T]$ gives the UMVUE. $\endgroup$ Commented Jul 6, 2017 at 1:03