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Given two Cauchy random variables $\theta_1 \sim \mathrm{Cauchy}(x_0^{(1)}, \gamma^{(1)})$ and $\theta_2 \sim \mathrm{Cauchy}(x_0^{(2)}, \gamma^{(2)})$. That are not independent. The dependence structure of random variables can often be quantified with their covariance or correlation coefficient. However, these Cauchy random variables have no moments. Thus, covariance and correlation do not exist.

Are there other ways of representing the dependence of the random variables? Is it possible to estimate those with Monte Carlo?

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Just because they don't have a covariance doesn't mean that the basic $x^t\Sigma^{-1} x$ structure usually associated with covariances can't be used. In fact, the multivariate ($k$-dimensional) Cauchy can be written as:

$$f({\mathbf x}; {\mathbf\mu},{\mathbf\Sigma}, k)= \frac{\Gamma\left(\frac{1+k}{2}\right)}{\Gamma(\frac{1}{2})\pi^{\frac{k}{2}}\left|{\mathbf\Sigma}\right|^{\frac{1}{2}}\left[1+({\mathbf x}-{\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x}-{\mathbf\mu})\right]^{\frac{1+k}{2}}} $$

which I have lifted from the Wikipedia page. This is just a multivariate Student-$t$ distribution with one degree of freedom.

For the purposes of developing intuition, I would just use the normalized off-diagonal elements of $\Sigma$ as if they were correlations, even though they are not. They reflect the strength of the linear relationship between the variables in a way very similar to that of a correlation; $\Sigma$ has to be positive definite symmetric; if $\Sigma$ is diagonal, the variates are independent, etc.

Maximum likelihood estimation of the parameters can be done using the E-M algorithm, which in this case is easily implemented. The log of the likelihood function is:

$$\mathcal{L}(\mu, \Sigma) = -{n\over 2}|\Sigma| - {k+1 \over 2}\sum_{i=1}^n\log(1+s_i)$$

where $s_i = (x_i-\mu)^T\Sigma^{-1}(x_i-\mu)$. Differentiating leads to the following simple expressions:

$$\mu = \sum w_ix_i/\sum w_i$$

$$\Sigma = {1 \over n}\sum w_i(x_i-\mu)(x_i-\mu)^T$$

$$w_i = (1+k)/(1+s_i)$$

The E-M algorithm just iterates over these three expressions, substituting the most recent estimates of all the parameters at each step.

For more on this, see Estimation Methods for the Multivariate t Distribution, Nadarajah and Kotz, 2008.

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  • $\begingroup$ That is a very good plan and a very detailed answer. One more question may be: Is it possible to write any joint Cauchy distribution like you did? For Gaussians, a similar answer is yes. But also for Gaussians correlation and dependence are equivalent. Is that also the case for Cauchy? $\endgroup$ – Jonas Jan 9 '19 at 7:04
  • $\begingroup$ Yes, this is the standard way of writing a multivariate Cauchy density. For the MV Cauchy, pseudo-correlation and dependence are also equivalent; all your intuitions carry over. $\sigma_{ij} = \sigma_i\sigma_j$ implies $x_i$ always $ = x_j$, etc. $\endgroup$ – jbowman Jan 9 '19 at 13:56
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While $\text{cov}(X,Y)$ does not exist, for a pair of variates with Cauchy marginals, $\text{cov}(\Phi(X),\Phi(Y))$ does exist for, e.g., bounded functions $\Phi(\cdot)$. Actually, the notion of covariance matrix is not well-suited to describe joint distributions in every setting, as it is not invariant under transformations.

Borrowing from the concept of copulas (which may also help in defining a joint distribution¹ for $(X,Y)$), one can turn $X$ and $Y$ into Uniform $(0,1)$ variates, by using their marginal cdfs, $\Phi_X(X)\sim\mathcal{U}(0,1)$ and $\Phi_Y(Y)\sim\mathcal{U}(0,1)$, and look at the covariance or correlation of the resulting variates.


¹For instance, when $X$ and $Y$ are both standard Cauchys,$$Z_X=\Phi^{-1}(\{\arg\tan(X)/\pi+1\}/2)$$is distributed as a standard Normal, and the joint distribution of $(Z_X,Z_Y)$ can be chosen to be a joint Normal $$(Z_X,Z_Y) \sim \mathcal{N}_2(0_2,\Sigma)$$This is a Gaussian copula.

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  • $\begingroup$ Thank you for your answer. I am not entirely sure though, whether this the right way to go. Values sampled with the Cauchy distribution will potentially be very large. When transforming them like this to a Gaussian, we probably end up putting all values in a very small set at the tail of the Gaussian. In which case, we can still estimate a covariance, but I guess the correlation would be close to 1. $\endgroup$ – Jonas Jan 9 '19 at 7:20
  • $\begingroup$ My point is that the correlation is a linear measure of dependence depending on the parametrerisation of the distribution, And once the two Cauchy variates are turned into Gaussians, their correlation can be anything between -1 and 1. Check the copula keyword on Wikipedia. $\endgroup$ – Xi'an Jan 9 '19 at 7:29

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