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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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1 Answers1

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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? 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