gl.check.lambda {gld}R Documentation

Function to check the validity of parameters of the generalized lambda distribution

Description

Checks the validity of parameters of the generalized lambda. In the case of the FMKL parameterisation, this is just a case of checking that lambda 2 is positive. The RS parameterisation is much more complex.

Usage

gl.check.lambda(lambda1, lambda2, lambda3, lambda4, parameterisation="fmkl")

Arguments

Note that the numbering of the lambda parameters is different to that used by Freimer, Mudholkar, Kollia and Lin (1988).

lambda1 lambda 1 - location parameter
lambda2 lambda 2 - scale parameter
lambda3 lambda 3 - first shape parameter
lambda4 lambda 4 - second shape parameter
parameterisation choose parameterisation: fmkl uses Freimer, Mudholkar, Kollia and Lin (1988) (default). rs uses Ramberg and Schmeiser (1974)

Details

See GeneralisedLambdaDistribution for details on the generalised lambda distribution. This function determines the validity of parameters of the distribution.

The FMKL parameterisation gives a valid statistical distribution for any real values of lambda 1, lambda 3,lambda 4 and any positive real values of lambda 2.

For the RS parameterisation, the combinations of parameters value that give valid distributions are the following (the region numbers in the table correspond to the labelling of the regions in Ramberg and Schmeiser (1974) and Karian, Dudewicz and McDonald (1996)):

region lambda 1 lambda 2 lambda 3 lambda 4 note
1 all <0 < -1 > 1
2 all <0 > 1 < -1
3 all >0 >= 0 >= 0 one of lambda 3 and lambda 4 must be non-zero
4 all <0 <= 0 <= 0 one of lambda 3 and lambda 4 must be non-zero
5 all <0 > -1 and < 0 >1 equation 1 below must also be satisfied
6 all <0 >1 > -1 and < 0 equation 2 below must also be satisfied

Equation 1

( (1-lambda3) ^ ( 1 - lambda3) * (lambda4 -1) ^ (lambda4 -1) ) / ( (lambda4 - lambda3) ^ (lambda4 - lambda3) ) < - lambda3 / lambda 4

Equation 2

( (1-lambda4) ^ ( 1 - lambda4) * (lambda3 -1) ^ (lambda3 -1) ) / ( (lambda3 - lambda4) ^ (lambda3 - lambda4) ) < - lambda4 / lambda 3

Note

The complex nature of the rules in this function for the RS parameterisation are the reason for the invention of the FMKL parameterisation and its status as the default parameterisation in the other generalized lambda functions.

Author(s)

Robert King, robert.king@mailbox.gu.edu.au, http://www.ens.gu.edu.au/robertk/

References

Freimer, M., Mudholkar, G. S., Kollia, G. & Lin, C. T. (1988), A study of the generalized tukey lambda family, Communications in Statistics - Theory and Methods 17, 3547–3567.

Karian, Z.E., Dudewicz, E.J., and McDonald, P. (1996), The extended generalized lambda distribution system for fitting distributions to data: history, completion of theory, tables, applications, the ``Final Word'' on Moment fits, Communications in Statistics - Simulation and Computation 25, 611–642.

Ramberg, J. S. & Schmeiser, B. W. (1974), An approximate method for generating asymmetric random variables, Communications of the ACM 17, 78–82.

http://www.ens.gu.edu.au/robertk/gld/

See Also

The generalized lambda functions GeneralisedLambdaDistribution

Examples

gl.check.lambda(0,1,.23,4.5) ## TRUE
gl.check.lambda(0,-1,.23,4.5) ## FALSE 
gl.check.lambda(0,1,0.5,-0.5,"rs") ## FALSE