# 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 having probability density function of the following form (1)

where is a fixed arbitrary number (more about it later), and (2)

denote the standard Normal (Gaussian) density function and its distribution function (the latter evalutated at point ), respectively. The component is called the shape parameter because it regulates the shape of the density function, as illustrated by some graphs having , and .

The density enjoys various interesting formal properties. It is easy to check that

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

The above random variable and its density function 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 (3)

which is then said to have a skew-normal distribution with parameters and write (4)

We refer to , and as the location, the scale and the shape parameters, respectively. Notice that, when , we re-obtain the 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 are as follows:
 mean value variance skewness kurtosis where .

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 (5)

irrespective of the value of .

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

• all its marginal components have a skew-normal distribution;
• its shape is regulated by a vector parameter ; when this is 0, we are back to the familiar multivariate normal distribution;
• various other typical properties of the multivariate normal distribution are preserved, in particular: (i) linear transformations of the form for any matrix are still multivariate skew-normal variates, and (ii) the distribution of certain quadratic forms is preserved.
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