## Approximating disbutions using the generalised lambda distribution

 Robert A. R. King H.L. MacGillivray Faculty of Environmental Science and Engineering, Griffith University, Brisbane, Australia and Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, Brisbane, Australia Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, Brisbane, Australia

## Outline of a paper presented at the Sydney International Statistical Congress

### Abstract

The generalised lambda distribution (g.l.d.) can take on many different shapes. It can be used to approximate all standard unimodal distributions. This paper presents approximations using both parameterisations of distribution - the original, published by Ramberg, Tadikamalla, Dudewicz and Mykytka [ Technometrics 21 (1979):78--82] and the parameterisation presented by Freimer, Mudholkar, Kollia and Lin [ Communciations in Statistics Theory and Methods 17 (1988):3547--67]. Approximation by the g.l.d. may be useful in easily producing quantiles or simulated data (as the g.l.d. is a transformation of the uniform). Two different methods for calculating the parameters for this approximation are presented. One of the methods provides the possibility of producing distributions that differ, to a desired degree, from a standard distribution in shape.

### Main points

1. We assume that our audience already thinks that it is useful to approximate some distributions with other ones, especially distirbutions like the gld which have F^{-1}(u) in closed form.
2. At least one of the methods gives a closer approximation than that provided by the method of moments
3. Both the starship and the quantile function methods allow the user to "tune" the method to take more account of a particular aspect of the approximation (for example the fit in the tails)
4. The quantile function method is based on a sounder appreciation of the effects of shape.

My page on the generalised lambda distribution gives the definition of the distribution and examples of its probability density function. It also features a java applet that draws probability density functions for the distribution.

### Overhead slides from the presentation

These are in postscript form:
• This has most of the Overheads for the presentation
```SISC.ps        386 Kb    Tue Jul 23 14:29:35 1996 Postscript Program
```
• This contains the table giving the parameter values for approximations to well-known distirbutions using the generalised lambda distribution
```partab.ps  57 Kb    Tue Jul 23 14:29:35 1996 Postscript Program
```
The table is also available as a html file with <pre>- formated text, or a plain text file.

File "rking/publ/biomet97.htm" last updated 06:53:41 AM, Fri Apr 21, 2006