Cramer-Rao lower bound for the variance of the unbiased estimator of $\tau(\theta)$

Question

Given the pdf $f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$ ; $-\inf < x<\inf$, $-\inf < \theta<\inf$

Show that the Cramer-Rao lower bound is 2/n

where n is the sample size.

Answer 1

1- you have the assumption that the samples are independent.

2- calculate the log likelihood ratio for $n$ samples. All likelihoods are multiplicative due to independence.

3- find the second derivative of the log likelihood function.

4- take the minus expectation w.r.t $X$ of what you've found at 3. And take the -1st power.

See http://en.m.wikipedia.org/wiki/Cramér–Rao_bound for more information