## Rao-Cramér bound

Hi all,

I have a question about the R-C bound for the variance of estimators of a parameter $\theta \in \Theta$.

Firstly: an efficient estimator attains its R-C bound, yes?

Secondly: in my course the lower bound is given as $\frac{1}{nI(\theta)}$ but elsewhere I have seen it given as $\frac{1}{I(\theta)}$ (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 $(1,\alpha)$ where

$f_{1,\alpha}(x) = e^{-(x-\alpha)}$

Show that (and explain why) the R-C bound does not hold for the efficient estimator $\alpha^{*} = X_{(1)}-1/n$.

I have shown that the variance of the estimator is equal to $1/n^2$ and I have shown that the Fisher information is equal to $n^2$. 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 !