A very brief introduction to the skew-normal distribution

This is a very short introduction to the Skew-Normal (SN) distribution. For references to the proper publications, available software and more information, see the SN home-page at http://azzalini.stat.unipd.it/SN

Consider first a continuous random variable $X$ having probability density function of the following form

f(x) = 2 \phi(x) \Phi(\alpha x)
\end{displaymath} (1)

where $\alpha$ is a fixed arbitrary number (more about it later), and
\phi(x)= \exp(-x^2/2)/\sqrt{2\pi} ,
\qquad \Phi(\alpha x) = \int_{-\infty}^{\alpha x}\phi(t) dt
\end{displaymath} (2)

denote the standard Normal (Gaussian) density function and its distribution function (the latter evalutated at point $\alpha x$), respectively. The component $\alpha$ is called the shape parameter because it regulates the shape of the density function, as illustrated by some graphs having \fbox{$\alpha=2$}, \fbox{$\alpha=5$} and \fbox{$\alpha=-5$}.

The density $f(x)$ enjoys various interesting formal properties. It is easy to check that

  1. when $\alpha=0$, the skewness vanishes, and we obtain the standard Normal density,
  2. as $\alpha$ increases (in absolute value), the skewness of the distribution increases,
  3. when $\alpha\to\infty$, the density converges to the so-called half-normal (or folded normal) density function;
  4. if the sign of $\alpha$ changes, the density is reflected on the opposite side of the vertical axis.

The above random variable $X$ and its density function $f(x)$ are the basic components of the construct, but for practical numerical work we need to add location and scale parameters. Consider then the linear transform

Y \: =\: \xi  +  \omega  X
\end{displaymath} (3)

which is then said to have a skew-normal distribution with parameters $ (\xi, \omega, \alpha), $ and write
Y\: \sim\: SN(\xi, \omega^2, \alpha).
\end{displaymath} (4)

We refer to $\xi$, $\omega$ and $\alpha$ as the location, the scale and the shape parameters, respectively. Notice that, when $\alpha=0$, we re-obtain the $N(\xi, \omega^2)$ distribution. You can see additional plots, varying all three parameters, with a demostration program which produces graphs of the univariate skew-normal densities.

Some characteristic values of the random variable $Y$ are as follows:
     mean value $\E{Y}= \xi+\omega\:\sqrt{2/\pi} \delta$
  variance $ \var{Y}= \omega^2\;(1-2 \delta^2/\pi)$
  skewness $ \gamma_1
= \dfrac{4-\pi}{2} \; \frac{\E{X}^3}{\var{X}^{3/2}} $
  kurtosis $ \gamma_2 = 2(\pi-3) \frac{\E{X}^4}{\var{X}^{2}} $
where $\delta=\alpha/\sqrt{1+\alpha^2}, \: \E{X}= \sqrt{2/\pi} \delta, \:
\var{X}= 1-2 \delta^2/\pi$.

On the statistical side, the skew-normal distribution is often useful to fit observed data with "normal-like" shape of the empirical distribution but with lack of symmetry. You can try it out directly with your data using a form available here.

Ok, but why is it called skew-normal? One reason is implicit in the above passage: this family of distributions includes the standard N(0,1) distribution as a special case, but in general its members have a skewed density. Another appealing connection is the fact that

X^2  \sim  \chi^2_1
\end{displaymath} (5)

irrespective of the value of $\alpha$.

So far for the univariate distribution; a multivariate version also exists. This refers to a multivariate variable $X$ such that

A demostration program which produces graphs of the bivariate skew-normal density allows to examine its shape for any given choice of the shape and association parameters.

The present account of the skew-normal distribution is clearly extremely limited. For an extended treatment, see the proper publications.

You may also wish to read a short historical note.

Back to the Skew-normal distribution

Adelchi Azzalini 2005-11-23