I have a question about the R-C bound for the variance of estimators of a parameter .
Firstly: an efficient estimator attains its R-C bound, yes?
Secondly: in my course the lower bound is given as but elsewhere I have seen it given as (for example, wikipedia). This is confusing as the definition of the Fisher information is the same.
What is the subtlety here? I ask because a problem I have been given is confusing me:
Suppose we are given the family of shifted exponential distributions with parameter where
Show that (and explain why) the R-C bound does not hold for the efficient estimator .
I have shown that the variance of the estimator is equal to and I have shown that the Fisher information is equal to . So if I use the version of R-C given on my lecture slides (btw: slide 141 here http://www.ms.unimelb.edu.au/~s620323/323_slides_7.pdf) it doesn't attain the R-C bound, so I can continue; but if I use the version which appears to be ubiquitous elsewhere, it actually does attain the bound.
Could somebody please explain this to me? Thanks !