Suppose that $X_1,X_2,\ldots,X_n$ form a random sample from a normal distribution with mean $0$ and unknown standard deviation $\sigma > 0$. Find the Cramer-Rao Lower Bound for the variance of any unbiased estimator of $\log \sigma$.

  • $\begingroup$ from my lecture-theory, we've used it for the more econometrics aspect of probability, looking at the most efficient estimators, the ones with the lowest variance. So we look at the paramater values such that cramer-rao lower bound is attained. $\endgroup$ – raditz Jun 1 '13 at 12:43
  • $\begingroup$ I think I have to find the fisher equation, but I have trouble getting it. $\endgroup$ – raditz Jun 1 '13 at 12:48

Note that the likelihood function is $f(\mathbb x|\sigma)=\left(\dfrac{1}{\sigma \sqrt{2\pi}}\right)^ne^{-\dfrac{1}{2}\dfrac{\Sigma x_i^2}{\sigma^2}}$.

Also find that $$\dfrac{\partial^2 \ln f(\mathbb x|\sigma) }{\partial \sigma^2}=\dfrac{n}{\sigma^2}-\dfrac{3}{\sigma^4}\Sigma x_i^2$$

So, Fisher's information is $I(\sigma)=E\left[ -\dfrac{\partial^2 \ln f(\mathbb x|\sigma) }{\partial \sigma^2}\right]=\dfrac{2n}{\sigma^2}$.

Here $\phi(\sigma)=\log \sigma$ so $\phi'(\sigma)=\dfrac 1{\sigma}.$

So $CRLB=\dfrac{(\phi'(\sigma))^2}{I(\sigma)}=\dfrac{1}{2n}$.

That's all from me.


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