Generalized Lambda distribution

The shape of generalized lambda distribution. Hold the button down to keep changing the value.
Generalized lambda distribution is a generic distribution that can be used for various curve fittings or in general mathematical analysis. It is interesting because of the wide variety of distributional shapes it can take on[1]. There are methods how to use this distribution to approximate various other distributions[2], or to fit experimental data set to this distribution[3].

There are two parameterisations of the distribution, both defined in the form of their inverse distribution function:

Ramberg and Schmeiser's

[[Math:c|F^{-1}(u)=\lambda_1 \plus {u^{\lambda_3} - (1-u)^{\lambda_4} } \over {\lambda_2}]]

Freimer, Mudholkar, Kollia and Lin's

[[Math:c|F^{-1}(u)=\lambda_1 \plus { { \frac{u^{\lambda_3} - 1}{\lambda_3} - \frac{(1-u)^{\lambda_4} - 1}{\lambda_4}} \over {\lambda_2}]]

Here lambda 1 is a location parameter, lambda 2 is a scale paramter and lambda 3 and lambda 4 together determine the shape.

[[Math:c|F^{-1}(u)]] for the Freimer case with the four lambdas equal to 0, 1, 0.13 and 0.13
[[Math:c|F^{-1}(u)]] for the Freimer case with the four lambdas equal to 0, 1, 0.13 and 0.13

These equations define a function [[Math:c|F^{-1}(u)]] where the parameter u must change between 0 and 1 (expected result of the function of the cumulative distribution). Inside this interval, the functions value makes the way from minus to plus infinity, representing that would be a parameter of the cumulative distribution function. Hence they are turned by 90 degree angle in comparison of the cumulative distribution that one may expect to see. The analytic form of the cumulative distribution function seem difficult to obtain in case of the generalized lambda distribution, and numeric methods must be used instead. The applet both "turns" the function in a proper orientation and computes the derivative, showing the approximate form of that the sample histogram would take.

The Generalised lambda Distribution is an extension of Tukey's lambda distribution. It was first suggested by first suggested Ramberg & Schmeiser in 1974.


  1. 1 Robert King's page at
  2. 2 R.A.R. King, H.L. MacGillivray (2000). Approximating disbutions using the generalised lambda distribution, includes source code in C
  3. 3 R.A.R. King, H.L. MacGillivray (1999). A Starship estimation method for the generalized lambda distribution, includes source code in C