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What are the formulaic parameter estimates for the skew-normal? If you can, the derivation via MLE or Mom would be great too. Thanks

Edit.

I have a set of data for which I can tell visually by plots is slightly skewed to the left. I want to estimate the mean and variance and then do a goodness-of-fit test (which is why I need the parameter estimates). Am I right in thinking I just have to guess the skew(alpha) (maybe do several skews and test for which is best?).

I would like the MLE derivation for my own understanding - would prefer MLE over MoM as I am more familiar with it.
I was unsure that there was more than one generic skew normal - I just mean a neg skewed mean! If possible, the skew exponential power param estimates would be helpful too!

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  • $\begingroup$ (1) which parameterization of which specific 'skew-normal'? (I've seen more than one thing get called that) (2) when you say "the formulaic parameter estimates" you imply (a) a closed form exists and (b) that there's only one --- yet you mention both ML and MoM, which won't generally be the same (& the ML estimators in particular might not be closed form). More information is required! $\endgroup$ – Glen_b Feb 17 '14 at 0:40
  • $\begingroup$ See, for instance, the paper by Vinod: Skew Densities and Ensemble Inference for Financial Economics, which illustrates how to fit data to a skew-Normal: mathematica-journal.com/issue/v9i4/SkewDensities.html $\endgroup$ – wolfies Apr 18 '14 at 7:04
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    $\begingroup$ In R, snormFit in fGarch will estimate a skew normal distribution, or you might prefer to look at the sn package (uses Azzalini's definition, beware that other definitions of "skew normal" exist). If you use Stata, try here. Various packages for Python, VBA and Perl are available from the site of Adelchi Azzalini at the University of Padua. $\endgroup$ – Silverfish Nov 26 '14 at 2:06
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Indeed, the "skew-normal family" has exploded in membership (the wikipedia article does not attest to this). So, let's consider the mother of them all, that has probability density function

$$f_X(x) = \frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha \left(\frac{x-\xi}{\omega}\right)\right)$$ where $\phi()$ is the standard normal pdf and $\Phi()$ the standard normal cdf. $\xi$ is the location parameter, $\omega$ is the scale parameter, and $\alpha$ is the skew parameter.

Closed-form solutions for the ML estimator do not exist. Method-of-Moments estimator provides closed forms as follows, assuming that all three parameters are non-zero (obviously if $\omega$ and/or $\xi$ are zero, then the steps below are simplified):

1) Obtain a MoM estimate $\hat \delta$ by solving for $\delta$ the expression for the skewness of the distribution,
enter image description here
using the estimated sample skewness coefficient $\hat \gamma_3$.

2) Obtain an estimate $\hat \alpha$ using

$$\delta = \frac {\alpha}{\sqrt {(1+\alpha^2)}} \implies \hat \alpha = \frac {\hat \delta}{\sqrt{1-\hat \delta^2}}$$

3) Obtain a MoM estimate $\hat \omega$ by solving for $\omega$ the expression for the variance, $$\hat \sigma^2_x = \omega^2\cdot \left(1-\frac{2\hat \delta^2}{\pi}\right)$$ using the sample variance and the estimated $\delta$ derived in the previous step

3) Obtain a MoM estimate $\hat \xi$ by solving for $\xi$ the expression for the mean of the distribution, $$\hat \mu_x = \xi + \hat \omega \hat \delta\sqrt {2/\pi}$$ using the sample mean and the previous estimates.

And don't forget to propagate estimation error in this sequential procedure, as regards the estimator variance.

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