The Skew-Normal Distribution

The Skew-Normal Probability Distribution

and related distributions, such as the skew-t and the SUN

The purpose of this page is to collect various material related to the Skew-Normal (SN) probability distribution and related distributions. The SN distribution is an extension of the normal (Gaussian) probability distribution, allowing for the presence of skewness.

A related famility is the skew-t (ST) distribution, which allows to regulate both skewness and kurthosis. The distribution is obtained by introducing a skewness parameter to the usual t density.

Other interesting parametric families belomng to the same borad formulations. Among them, a mention is due for the skew-exponential power (i.e. Subbotin) distibution and for the closed/unified skew-normal distribution (CSN/SUN).


Papers, bibliography, etc.

A monograph
A monograph entitled The Skew-Normal and Related Families has appeared on 2013-12-19. However, notice that the official publication year, as indicated in the publication data, is 2014. A paperback edition is also available (2018-05-11)

A detailed bibliography is available. Current update: 2022-02-09.

My own papers
Abstracts (some with full text) of my research articles in this area. (Updated 2021-11-26)

Some talks
Handouts of a selection of talks. (Updated 2018-10-22)
One of these consitutes A one-day course on symmetry-modulated distributions, complemented by An introduction to the R package 'sn'.

A pioneer
In 1908, Fernando de Helguero presented a paper which examines a selection mechanism of a normal population as a model of departure from normality. This construction essentially perturbates the normal density via a uniform distribution function, leading to a form of skew-normal density. Although mathematically somewhat different from the above-described form of skew-normal density, the underlying stochastic mechanism is intimately related. (2004-12-13)

Software: 'sn' package

The 'sn' package (or library, here the term is used as a synonym) is a suite of functions for handling skew-normal distribution and related ones (such as the skew-t and the CSN/SUN), in the univariate and the multivariate case. The available facilities include various standard operations (density function, random number generation, etc), data fitting via MLE, plotting log-likelihood surfaces and others. For data fitting, simple random samples and regression models are dealth with, for univariate and multivariate SN and ST error distributions.

Current development takes place in the R computing environment. Some porting to other languages are available but they are not really maintained: if you want the most recent version, use the one for R. Notice that most of the existing portings to other environments have been carried out before version 0.3-0, and therefore they do not include many facilities, for instance those for the skew-t distribution.

You can get the software from here.

Software: other facilities

For random numbers generation, specifically:

On-line procedures

Data fitting
You can fit a skew-normal distribution to your data using this form. This procedure also serves as a demonstration of the library sn functionality, although only in a simple case. If you have a more complex problem (large data set, data with covariates, multivariate data, etc), then you must download the full library and run it yourself. (Created on 2003-02-17, updated 2003-04-22, 2008-12-02, 2011-08-03).

Random numbers generation
You can generate random numbers with SN or ST distribution in 1 or 2 dimensions using this form (2003-11-12). See also the FAQ below.

Some recurring questions

Random numbers
How to generate random variates with SN or ST distribution? (2003-11-12).
R users (and possibly also users of something-else) may want to look at this document (update 2016-02-12)

About history
Where did the skew-normal distribution appear first? (2009-11-17)

Data from the Australian Institute of Sport (AIS)
Where do I find the AIS data? (2013-10-08)

A less frequent question
In the multivariate case, the feasible region for the set of correlations and the indices of skewness of the individual components is not simple to perceive. To help visualizing this region in the bivariate case, you can run the R program feasible-CP2.R; besides R, it requires its package 'rgl'. To run it, save this file locally, then start R and type source('feasible-CP2.R'). (2009-05-27)

The program displays two plots in sequence. The first plot adopts delta as the shape parameter; the connection between delta and gamma1 is described in various articles, including this one. The second plot uses gamma1.


Real SN distribution
The skew-normal distribution is not a mere mathematical abstraction: it is real life! (2013-06-06)

Translations of the term "skew-normal distribution" available at ISI

A research problem
The paper Statistical applications of the multivariate skew-normal distribution includes the discussion of an apparently innocuous dataset, but having the MLE on the frontier of the parameter space. Can you suggest an explanation of the phenomenon, and/or propose an alternative, `reasonable' estimate? It should work with this as well as with more regular datasets. Hence, the obvious answer (the method of moments) is not acceptable, since it would work here but not with other datasets having the sample index of skewness outside the feasible region. Various solutions to the problem have been put forward, both in the classical and in the Bayesian approach. You can get the `frontier' data, and try out your own method.

This page is updated from time to time; see the 'last update' date below.
Feedback of (almost) any sort is welcome. E-Mail: email

Note: the dates on this page are (mostly) in ISO-8601 format
Page created on 1998-10-12, last update on 2023-04-04 (this could reflect changes in some sub-page).

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