## 24 January 2016

### Sample median for a Lorentz (Cauchy) distribution - part 2

In a previous post I derived the distribution function for the median of a sample of size $$n = 2k +1$$ (with $$k \geq 0$$ integer) drawn from a Lorentz (or Cauchy) distribution:

\label{eq:result}
g(x) = \frac{n!}{(k!)^2 \, \pi ^{n} \, \gamma} \left [ \frac{\pi ^2}{4} - \arctan ^2 \left (\frac{x - x_0}{\gamma} \right )\right ]^k \frac{1}{1+ \left (\frac{x - x_0}{\gamma} \right )^2 }
I will now consider some of its properties.

The Figure below shows $$g(x)$$ with $$x_0 = 0$$ and $$\gamma = 1$$ for a few values of $$n$$:

As the sample size increases, the distribution is more and more localized, as its tails decay faster. The Lorentzian already behaves as $$x^{-2}$$ at infinity, and the parenthesis in front of it adds a factor $$x^{-k}$$. For $$k \geq 1$$ the distribution has a first moment: $$\left \langle x \right \rangle = x_0$$, so the median is an unbiased estimator of the position parameter $$x_0$$. For $$k \geq 2$$ it also has a variance $$V = \left \langle (x - x_0)^2 \right \rangle$$, which quantifies the tightness of the estimation [1]. Following Rider, I'll use the substitution $$\phi = \text{arccot}\left ( \frac{x - x_0}{\gamma}\right)$$, yielding:

\begin{eqnarray}
\label{eq:integ}
V &= \frac{n!}{(k!)^2 \, \pi ^{n} \gamma} \int _{0}^{\pi} \, \mathrm{d} \phi \, \gamma \, (1+ \cot ^2 \phi) \left [ \frac{\pi}{2} - \arctan \left (\frac{x - x_0}{\gamma} \right )\right ]^k \nonumber \\
& \left [ \frac{\pi}{2} + \arctan \left (\frac{x - x_0}{\gamma} \right )\right ]^k \frac{\gamma ^2 \cot ^2 \phi}{1+ \cot ^2 \phi} \nonumber \\
&= \gamma ^2 \frac{2 n!}{(k!)^2 \, \pi ^{n}} \int _{0}^{\pi /2} \, \mathrm{d} \phi \, \phi ^k (\pi - \phi)^k \cot ^2 \phi = \gamma ^2 I_k
\end{eqnarray}
where $$I_k \sim \frac{8}{3 n}$$ can be obtained by numerical integration and the standard deviation of the median $$\sqrt{V}$$ decreases as $$1/\sqrt{n}$$, as per the central limit theorem.
TO DO: I found the behaviour of $$I_k$$ by inspecting the list of numerical values; try to get a more rigorous result. Compare with the large-$$n$$ limit $$\frac{\pi^2}{4 n}$$.

[1] P. R. Rider, Variance of the Median of Samples from a Cauchy Distribution. Journal of the American Statistical Association 55, 322–323 (1960).