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Considering the class of unbiased estimator:

If an estimator is efficient then it is an umvue

But if an estimator is an umvue it is not true in general that it is also effcient

What hypothesis is this second statement lacking in order to make it true?

my guess: the umvue has the minimum variance but only uniformly, while efficiency means that has the minimum variance more generally.

But then what generally would mean?

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If an UMVU estimator does not reach the Rao-Cramer lower bound, it is not efficient.

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  • $\begingroup$ Can you make a simple example on how it cannot reach the c.r. Lower bound? $\endgroup$
    – Lex
    Commented Aug 15, 2015 at 13:59
  • $\begingroup$ Also if the umvue is not efficient, then would it mean that an efficient estimator does not exist (inside or outside the class of unbiased estimator)? $\endgroup$
    – Lex
    Commented Aug 15, 2015 at 14:07
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    $\begingroup$ The classic example is for the variance in the normal distribution. The sample variance is the umvu estimator, but it is not efficient. $\endgroup$
    – mandata
    Commented Aug 15, 2015 at 14:15
  • $\begingroup$ to your second comment, yes, $\endgroup$
    – mandata
    Commented Aug 15, 2015 at 14:17
  • $\begingroup$ Are you sure the C-R lower bound always can be reached? $\endgroup$ Commented May 11, 2022 at 0:41

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