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
Consider first a continuous random variable having probability
density function of the following form
where is a fixed arbitrary number
(more about it later), and
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
The density enjoys various interesting formal properties.
It is easy to check that
- when , the skewness vanishes, and we obtain the
standard Normal density,
- as increases (in absolute value), the skewness of the
, the density converges to the so-called
half-normal (or folded normal) density function;
- 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
which is then said to have a skew-normal distribution with
We refer to
as the location, the scale and the shape parameters, respectively.
Notice that, when , we re-obtain the
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:
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
irrespective of the value of .
So far for the univariate distribution; a multivariate
version also exists. This refers to a multivariate variable
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.
- 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
- 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.
The present account of the skew-normal distribution is clearly
extremely limited. For an extended treatment, see the
You may also wish to read a short historical
Back to the Skew-normal distribution