# Maximum likelihood estimation for Cauchy noise

What is the maximum likelihood estimator of the covariance matrix for a given vector in the presence of Cauchy noise?

How can we calculate it given that the Cauchy distribution has infinite variance?

• How do you define a multivariate Cauchy distribution? Do you mean a special multivariate t? In which case the scale matrix is well-defined if not necessarily a covariance matrix. Commented Jul 20, 2015 at 8:43
• How does your question differ from that one? Commented Jul 20, 2015 at 8:45
• I am not using multivariate Cauchy distribution. I am using standard Cauchy Distribution. Commented Jul 20, 2015 at 8:55
• The multivariate Cauchy is a special case of the multivariate t with one degree of freedom. Commented Jul 20, 2015 at 9:42

As explained in Nadarajah and Kotz (2007), given the log-likelihood function of the multivariate t distribution with parameters $(μ,R,ν)$, $$L(μ,R,ν)=−\frac{n}{2}\log|R|−\frac{ν+p}{2}\sum_{i=1}^n\log(ν+s_i)\,,$$ the maximum likelihood estimator can be found by an EM algorithm exploiting the latent Gaussian representation of the t.
The EM iteration is of the form $$μ^{(m+1)}=\text{average}(w^{(m)}_ix_i)\big/\text{average}(w^{(m)}_i)$$ and $$R^{(m+1)}=\text{average}(w^{(m)}_i\{x_i-μ^{(m+1)}\}\{x_i-μ^{(m+1)}\}^\text{T})\big/\text{average}(w^{(m)}_i)$$ where $$w^{(m)}_i=(ν+p)\big/\{\nu+(x_i−μ^{(m)})(^\text{T}R^{(m)})^{-1}(x_i−μ^{(m)})\}$$