gl.check.lambda {gld} | R Documentation |
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.
gl.check.lambda(lambda1, lambda2, lambda3, lambda4, parameterisation="fmkl")
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) |
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
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.
Robert King, robert.king@mailbox.gu.edu.au, http://www.ens.gu.edu.au/robertk/
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, 35473567.
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, 611642.
Ramberg, J. S. & Schmeiser, B. W. (1974), An approximate method for generating asymmetric random variables, Communications of the ACM 17, 7882.
http://www.ens.gu.edu.au/robertk/gld/
The generalized lambda functions GeneralisedLambdaDistribution
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