The Skew-Normal Probability Distribution

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).

Introduction

• If you have never read about the skew-normal probability distribution, you may want to look at a very brief account.
• To view the shape of the density function, here are some graphical demostration programs:
  • plot of the univariate SN density;
  • plot of the bivariate SN density (1998-10-20);
  • a nice animation developed by Colin Rose shows the behaviour of the univariate SN density as the shape parameter moves along its range;
  • plot the univariate skew-t (ST) and skew-Cauchy density function. (2002-07-27)
  • plot the bivariate skew-t (ST) and skew-Cauchy density function (2003-07-02).

Publications, 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)

Bibliography

A detailed bibliography is available. Current update: 2023-09-29. Note: due to the increasing number of pertaining publications and the difficulty in keeping track of them, this may be the final update of this bibliography.

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

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.

• The key edition of the package is the one for R. Package version 2.1.1 (2023-04-04) > is available from here and from CRAN. You can look at an overview of the package (2017-02-22), at a tutorial introduction (2017-10-10) and its manual.
• The upgrade from version 0.4-18 to the 1.x-y series has implied that the new version is only partly backward compatible with the 0.4-xx version. For those who want to keep both versions, a modified package called sn0 has been created. This is simply a rename of the previous version (with renaming of some internal references) and is available from here. It has been packaged for Unix, but it should work with MS-windows as well since no binaries are used. (2014-01-07).
• Notice that the R versions of 'sn' for various platforms are also obtainable directly from CRAN, and they are simply installed using the install.packages("sn") command, provided that your installation is suitably configured (and that you are actually connected on-line!).
• The library has been ported to Matlab by Nicola Sartori. However, this porting refers to update 0.21 of the package, that is, to year 2000! Hence many facilities are not included, notably those for the skew-t distribution. A subset of the facilities for the skew-t distribution is available via a set of Matlab functions which have been written and made available by Enrique Batiz (Enrique.Batiz [at] postgrad.mbs.ac.uk). Please see the note for Matlab users (2013-12-11).
• An MS-DOS executable program is available which implements a small portion of the `sn library', namely MLE estimation for random samples and for regression models with errors having scalar SN distribution.
• The above-described executable program is based on some Fortran90 code which represents the porting of a few basic routines from R. About 10 years after writing it, prompted by a user request, I have thought that after all this Fortran code could be of use to other people; so now it is available. (Written in 1998, on line since 2008-09-25)

Software: other facilities

• The R package gamlss and related ones allow use of skew-normal and connected distributions; see also gamlss.dist and the GAMLSS web page
• The R package VGAM includes treatment of the skew-normal distribution
• Boost C++ routines are available within the Math Toolkit
• MathStatica 1.5
• NIST-Dataplot (in Chapter 8) includes some functions for handling skew-normal/skew-t distributions, namely

    SNCDF/PDF/PPF (Skew-Normal CDF/PDF/PPF)
    STCDF/PDF/PPF (Skew-t CDF/PDF/PPF)
    LSNCDF/PDF/PPF (Log-Skew-Normal CDF/PDF/PPF)
    LSTCDF/PDF/PPF (Log-Skew-t CDF/PDF/PPF)     

• Routines in Gauss written by Thierry Roncalli (2008-10-21)
• An additional set of R routines is available for handling ESN (Extended SN) distributions, written by Adelchi Azzalini years ago, but posted here much later (2011-03-08). The software is freely usable under the conditions indicated inside the file itself.
• For Python:
  • a set of Python routines for the univariate skew-normal variates has been made available by Janwillem van Dijk (2013-01-29);
  • additional functions are documented here, along with access to the source code (2020-12-31).
• For Stata users there is A suite of commands for fitting the skew-normal and skew-t models

• Excel users can make use of VBA routines kindly made available by Stephen H. Gersuk (2008-09-22);
• a Perl module has been provided by Jiri Vaclavik (added 2011-10-21);

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.

Miscellanea

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:

Note: the dates on this page are (mostly) in ISO-8601 format
Page created on 1998-10-12, last update on 2023-09-29 (this could reflect changes in some sub-page).