The Generalised Lambda Distribution

also known as the Asymmetric Lambda Distribution


The Generalised λ (lambda) Distribution is an extension, first suggested by Ramberg & Schmeiser (1974), of Tukey's λ distribution.

There are two parameterisations of the distribution, both defined by their inverse distribution function. The two inverse distribution functions are here, labelled by the paper that first introduced them.

Ramberg and Schmeiser's (LaTeX format) F^{-1}(u)= \lambda_1 + {u^{\lambda_3} - (1-u)^{\lambda_4}
Freimer, Mudholkar, Kollia and Lin's*  (LaTeX format) F^{-1}(u)=
F^{-1}(u)= \lambda_1 + { { \frac{u^{\lambda_3}}{\lambda_3} -
\frac{(1-u)^{\lambda_4}}{\lambda_4} } \over \lambda_2 }

Here λ1 (lambda 1) is a location parameter, λ2 is an inverse scale parameter and λ3 and λ4 jointly determine the shape (with λ3 mostly effecting the left tail and λ4 mostly effecting the right tail).

It is of interest because of the wide variety of distributional shapes that it can take on. The java applet on this page shows the probability density function for any combination of parameters.

You need to click in the graph image to get it to update the graph

Couldn't present the applet
The source code for DisplayLambda is available to you under the GPL.

Note (parameter labels for FMKL parameterisation)

Note that this labels the parameters differently from Freimer, Mudholkar, Kollia and Lin's original paper (which labelled the shape parameters as lambda 1 and lambda 2). The renaming is made to make it consistent with Ramberg and Schmeiser.

The source code for DisplayLambda is available to you under the GPL.
Here are some examples of the density shapes produced by the gld:

4 gld probability density function graphs

In case you have difficulties reading the titles on the graphs, they are graphs of the probability density function for the generalised lambda distribution (Ramberg's parameterisation) for the following parameter values: (0,1,.1,.1), (0,1,.05,.5), (0,-5,-.05,-.1) and (0,1,1.4,1.9)

See my publications information section for information on articles and conference papers related to this distribution.

I have written an add-on package for R that provides this distribution. More information is available here.


Robert King's page
Statistics
Mathematical and Physical Sciences
University of Newcastle
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File "public_html/gld/index.html" last updated 07:59:13 AM, Wed Apr 05, 2006
comments to: robert.king@newcastle.edu.au