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:

$$\begin{equation} \label{eq:result} g(x) = \frac{n!}{(k!)^2 \, \pi ^{n} \, \gamma} \left [ \frac{\pi ^2}{4} - \arctan ^2 \left (\frac{x - x_0}{\gamma} \right )\right ]^k \frac{1}{1+ \left (\frac{x - x_0}{\gamma} \right )^2 } \end{equation}$$

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 bracket 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 $\frac{\pi^2}{4 n}$.

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