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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?

Xi'an
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undefined
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    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. – Xi'an Jul 20 '15 at 08:43
  • How does your question differ from [that one](http://stats.stackexchange.com/q/125381/7224)? – Xi'an Jul 20 '15 at 08:45
  • I am not using multivariate Cauchy distribution. I am using standard Cauchy Distribution. – undefined Jul 20 '15 at 08:55
  • The multivariate Cauchy is a special case of the multivariate ***t*** with one degree of freedom. – Xi'an Jul 20 '15 at 09:42

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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 $$w^{(m)}_i=(ν+p)\big/\{\nu+(x_i−μ^{(m)})(^\text{T}R^{(m)})^{-1}(x_i−μ^{(m)})\}$$

Xi'an
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  • Thanks but as this is for multivariate t distribution as I need for cauchy me degree of freedom will be 1. We will get MLE of multivariate Cauchy. – undefined Jul 20 '15 at 09:23
  • Okay what will be our first assumption to start iteration. – undefined Jul 20 '15 at 10:07
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    Have you checked the basics of [the EM algorithm](https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm)? The starting point only matters in determining the limiting value of the algorithm and should thus be modified for multiple runs of EM. – Xi'an Jul 20 '15 at 12:01
  • Can you tell me what will be the distribution of the ML covariance matrix for a Cauchy noise? – undefined Aug 03 '15 at 05:16
  • Sorry sir, your answers are really helping me to learn, I am new to use stack exchange and a new learner may be due to this I missed but I found most of your answers useful. – undefined Aug 03 '15 at 08:50