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
Freimer, Mudholkar, Kollia and Lin's*

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

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.

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