# How to calculate the scale parameter of a Cauchy random variable

Let $(X_n)$ be iid random variables and suppose they have mean 0 and follow Cauchy distribution.

I know I can set the location parameter to 0. My question is how to find the corresponding scale parameter. Thank you!

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Note that the mean of the Cauchy distribution doesn't exist. However, I assume you mean the center of symmetry of the Cauchy (which is both the mode and the median and various other measures of location).

Let's call the center $\mu$ and the scale parameter $\sigma\,$:

$$f(x; \mu,\sigma) = \frac{1}{\pi\sigma \left[1 + \left(\frac{x - \mu}{\sigma}\right)^2\right]} ,\quad \sigma>0,\;\;x,\mu\in \mathbb{R}$$

1. Quick estimate: A reasonable quick estimate for $\sigma$ can be obtained from half the interquartile range. This ignores that $\mu$ is known, of course. The median absolute value would be a corresponding quantity for $\mu=0$. If memory serves, I think the asymptotic relative efficiency (ARE) is about 80% for that median absolute value, but don't quote me on that.

2. Maximum likelihood: Let $X$ be $\sim\text{Cauchy}(\mu_0,\sigma)$ for known $\mu_0$.

The MLE for $\sigma$ is given by solving the following for $\hat\sigma\,$:

$$\sum_i \frac{\hat{\sigma}^2}{(x_i-\mu_0)^2+\hat{\sigma}^2}=\frac{n}{2}\,.$$

A solution exists for any $n>2$ and is unique (e.g. see Copas, 1975$^{[2]}$). This performs well, but must be iterated.

3. Efficient one-step estimation: This$^{[2]}$ recent paper gives an efficient (ARE ~98%) and simple estimate based on the Hodges-Lehmann estimator, as well as some useful details on the ML estimator. In particular, section 3 gives details for the known-location case:

When the known location is zero, they show that half of the median of $n(n+1)/2$ logarithms of the absolute values of pairwise products of the Cauchy observations is ML for $\log(\sigma)$.

That is, $\log(\hat{σ}_\text{HLE}) =\frac{1}{2}\text{med}(\ln|X_iX_j|), 1≤i, j≤n, i≤j$.

This is unbiased for $\log(\sigma)$ and asymptotically normally distributed. As a result, exponentiating that is a suitable (though not unbiased) estimator for $\sigma$. Further details are in the paper, including the variance of the asymptotic normal distribution for the estimator of $\log(\sigma)$.

Rousseeuw and Croux (1993)$^{[3]}$ also give a robust scale estimator, $Q_n$ (for the unknown location case) which is also highly efficient at the Cauchy. It's based on scaling the first quartile of the pairwise absolute distances between observations. (It's available in the package robustbase in R, but for the Cauchy you need a different scale factor than the default scaling constant.)

References

$[1]$ Copas, J.B., (1975),
"On the unimodality of the likelihood for the Cauchy distribution,"
Biometrika, 62(3):701-04.

$[2]$ Kravchuk, O.Y. and Pollett, P.K. (2012),
"Hodges-Lehmann scale estimator for Cauchy distribution."
Communications in Statistics-Theory and Methods, Vol 41(20):3621-3632.

(See also this version of the paper given at the web page of the second author, here)

$[3]$ Rousseeuw, P. and Croux, C. (1993),
"Alternatives to the median absolute deviation,"
Journal of the American Statistical Association, 88(424):1273-1283

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