How can I solve the MLE for the skew-t distribution via EM? I am comfortable with the EM methods for t, so could someone show it for the skew-t?

  • 1
    $\begingroup$ There are multiple definitions of the skew-$t$ distribution so you need to start by specifying which one you are using. However, my suggestion is the same as for your previous question - skip the EM algorithm and use numerical maximization of the likelihood. $\endgroup$
    – M. Berk
    Feb 21, 2014 at 15:35
  • $\begingroup$ Do you mean specifying skew? The problem is I don't know the skew yet. I had hoped to be able to plug arbitrary values for the skew parameter and graph it to find the maximum - but this might not be possible with 4 parameters $\endgroup$
    – user40124
    Feb 21, 2014 at 15:54
  • $\begingroup$ I mean how the density of the distribution is derived. I'll write up a derivation for one possible definition $\endgroup$
    – M. Berk
    Feb 21, 2014 at 16:01

1 Answer 1


I will base my answer on the skew-$t$-normal distribution of Gomez et al (2007) (see the end of the answer for full references) for reasons discussed shortly. Let $z \sim StN(\mu,\sigma^2,\lambda,\nu)$ denote that $z$ follows a skew-$t$-normal distribution with location parameter $\mu$, scale parameter $\sigma^2$, skewness parameter $\lambda$ and degrees of freedom parameter $\nu$. Then the density of $z$ is given by:

$$ f(z|\mu,\sigma^2,\lambda,\nu) = 2t_\nu(z|\mu,\sigma^2)\Phi\left(\frac{z-\mu}{\sigma}\lambda\right) $$

where $t_\nu(z|\mu,\sigma^2)$ denotes the $t$ density with $\nu$ degrees of freedom, location parameter $\mu$ and scale parameter $\sigma^2$. $\Phi$ denotes the Gaussian cumulative distribution function.

This form is identical to the skew-$t$ distribution proposed by Azzalini and Capitanio (2003) except that the Gaussian CDF controlling the skewness of the density is replaced by the Student $t$ CDF, which depends not just on $\lambda$ but also the degrees of freedom parameter $\nu$. The form given here has the advantage of being computationally simpler to evaluate and I think it's better to decouple the skewness from the degrees of freedom parameter.

To proceed with the EM algorithm, note the following hierarchical decomposition:

$$ \tau \sim \Gamma\left(\frac{\nu}{2},\frac{\nu}{2}\right)\\ \gamma|\tau \sim TN\left(0,\frac{\tau+\lambda^2}{\tau};(0,\infty)\right)\\ z|\tau,\gamma \sim N\left(\mu + \frac{\sigma\lambda}{\tau + \lambda^2}\gamma,\frac{\sigma^2}{\tau + \lambda^2}\right) $$

Where $TN(\mu,\sigma^2;(a,b))$ denotes the truncated normal distribution lying within the interval $(a,b)$. So the idea is to treat $\tau$ and $\gamma$ as latent variables. These are the maximum likelihood estimators of the complete log likelihood:

$$ \widehat{\mu} = \frac{\sum_{i=1}^n[\tau_iz_i] + \lambda^2\sum_{i=1}^nz_i-\lambda\sigma\sum_{i=1}^n\gamma_i}{n\lambda^2+\sum_{i=1}^n\tau_i}\\ \widehat{\sigma^2}=\frac{1}{n}\sum_{i=1}^n\tau_i(z_i-\mu)^2\\ \widehat{\delta}=\frac{\sum_{i=1}^n(z_i-\mu)\gamma_i}{\sum_{i=1}^n(z_i-\mu)^2}\\ \widehat{\lambda}=\widehat{\delta}\widehat{\sigma} $$

there is no closed form estimator $\nu$ so instead you will have to numerically maximize:

$$ \frac{\nu}{2}\log\frac{\nu}{2} - \log\Gamma(\frac{\nu}{2}) + \frac{\nu-1}{2}\log\tau_i-\frac{\nu}{2}\tau_i $$

These are the conditional expectations of the latent variables:

$$ E[\gamma_i|z_i] = (z_i - \mu)\frac{\lambda}{\sigma} + \frac{\phi\left((z_i - \mu)\frac{\lambda}{\sigma}\right)}{\Phi\left((z_i - \mu)\frac{\lambda}{\sigma}\right)}\\ E[\tau_i|z_i] = \frac{\nu+1}{\nu + \frac{(z_i - \mu)^2}{\sigma^2}} $$

I apologize in advance for any potential mistakes in the formulae and suggest you verify them by deriving them yourself.

Also, as I said in my comment, I believe you are better off foregoing the EM algorithm and simply maximizing the observed likelihood with, for example, Nelder-Mead.


Azzalini, Adelchi, and Antonella Capitanio. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution." Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65, no. 2 (2003): 367-389.

Gómez, Héctor W., Osvaldo Venegas, and Heleno Bolfarine. "Skew‐symmetric distributions generated by the distribution function of the normal distribution." Environmetrics 18, no. 4 (2007): 395-407.

  • 1
    $\begingroup$ +1. Can you also post the full bibliographic references to Gomez et al (2007) and Azzalini and Capitanio (2003)? $\endgroup$
    – user603
    Feb 21, 2014 at 20:57
  • 1
    $\begingroup$ @user603 done. Sadly Gomez et al (2007) is not available without subscription. $\endgroup$
    – M. Berk
    Feb 21, 2014 at 22:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.