I am tasked with determining if a particular parameter can be estimated efficiently.

Given that an efficient estimator is an unbiased estimator which achieves the Cramer-Rao lower-bound, is the only way to answer this question by first finding the UMVUE, calculating its variance, and then comparing its variance to the reciprocal of the Fisher information in the sample? Sounds like a whole lot of calculation, not that it should be too difficult, but I'm wondering if there's a shortcut.

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    $\begingroup$ Unless you can invoke the Lehmann-Scheffé theorem, which avoids the derivation of the Cramer-Rao lower-bound. $\endgroup$
    – Xi'an
    Jan 16 '19 at 5:49
  • $\begingroup$ Thanks for the comment. So let's say that I use Lehmann-Scheffe to identify a UMVUE. Wouldn't I still have to calculate its variance, and calculate the Cramer-Rao lower-bound? I'm not sure what you mean when you say that using Lehmann-Scheffe is a way of avoiding the derivation of the CRLB. $\endgroup$ Jan 16 '19 at 13:54
  • $\begingroup$ Then how would I know if the statistic achieves the CRLB, i.e. if the estimator is efficient? The UMVUE is not necessarily efficient. Perhaps I'm not understanding something... $\endgroup$ Jan 16 '19 at 14:47
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    $\begingroup$ I have a rather lax approach to efficiency! Being minimum variance among unbiased estimators is a form of efficiency. Now, to quote from Wikipedia, "an efficient estimator need not exist, but if it does and if it is unbiased, it is the UMVUE." $\endgroup$
    – Xi'an
    Jan 16 '19 at 20:47
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    $\begingroup$ Except for general results about exponential families, I have no idea. Just that in the majority of cases, there is no efficient estimator, as illustrated with the James-Stein paradox. $\endgroup$
    – Xi'an
    Jan 16 '19 at 21:02

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