Discussion
Cramer-Rao lower bound of variance for estimating the parameter \(\theta\) of the distribution with pdf :
\[ f(x,\theta)=\frac{1}{\pi}\,\frac{1}{1+(x-\theta)^{2}} \]
where \(-\infty \lt x \lt \infty\) is :
Answer: A
The Cramer-Rao lower bound is 2/n.
For the Cauchy location family, \(\log f = -\log\pi - \log(1 + (x - \theta)^{2})\), and the Fisher information in one observation is
\[ \begin{aligned} I(\theta) &= E\left[\frac{4(X - \theta)^{2}}{(1 + (X - \theta)^{2})^{2}}\right] \\ &= \frac{1}{2} \end{aligned} \]
(the integral \(\frac{4}{\pi}\int \frac{u^{2}}{(1 + u^{2})^{3}}du\) equals \(\frac{1}{2}\)). For a sample of n, the information is \(n/2\), so the bound for an unbiased estimator of \(\theta\) is \(1/(n/2) = 2/n\).
1/n would correspond to information 1 per observation, and 1/(2n) inverts the factor 2. \(2/n^{2}\) has the wrong power of n.
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