Discussion
Consider the following statements :
(I) The Fisher information \(I(\theta)\) associated with a logistic distribution with location parameter \(\theta\) and scale parameter 1 is 1/3.
(II) The Fisher information \(I(\theta)\) associated with a Cauchy distribution with location parameter \(\theta\) and scale parameter 1 is 1/2.
Answer: C
Both (I) and (II) are true.
For a location family \(f(x - \theta)\), the Fisher information is \(I = \int (f'/f)^{2} f\, dx\), the same for every \(\theta\).
(I) For the standard logistic, \(f'/f = -(1 - 2F(x))\), and \(1 - 2F\) is uniform on (-1, 1) when X is logistic. So \(I = E[(1 - 2F)^{2}] = \int_{-1}^{1} \frac{u^{2}}{2}\, du = \frac{1}{3}\).
(II) For the standard Cauchy, \(f'/f = -2x/(1 + x^{2})\), and
\[ \begin{aligned} I &= \int_{-\infty}^{\infty} \frac{4x^{2}}{(1 + x^{2})^{2}} \cdot \frac{dx}{\pi(1 + x^{2})} \\ &= \frac{1}{2} \end{aligned} \]
Both values are standard results, so the options denying either statement are wrong.
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