Function to check the validity of parameters of the generalized lambda distribution
Checks the validity of parameters of the generalized lambda. The tests are simple for the FMKL, FM5 and GPD types, and much more complex for the RS parameterisation.
gl.check.lambda(lambdas, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, param = "fkml", lambda5 = NULL, vect = FALSE)
lambdas |
This can be either a single numeric value or a vector. If it is a vector, it must be of length 4 for parameterisations
If it is a a single value, it is lambda 1, the location parameter of the distribution and the other parameters are given by the following arguments Note that the numbering of the lambda parameters for the fmkl parameterisation is different to that used by Freimer, Mudholkar, Kollia and Lin. |
lambda2 |
lambda 2 - scale parameter (β for |
lambda3 |
lambda 3 - first shape parameter (δ, skewness parameter for |
lambda4 |
lambda 4 - second shape parameter (λ, kurtosis parameter for |
lambda5 |
lambda 5 - a skewing parameter, in the fm5 parameterisation |
param |
choose parameterisation:
|
vect |
A logical, set this to TRUE if the parameters are given in the
vector form (it turns off checking of the format of |
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.
The FM5 parameterisation gives statistical distribution for any real values of lambda 1, lambda 3, lambda 4, any positive real values of lambda 2 and values of lambda 5 that satisfy -1 <= lambda5 <= 1.
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 GPD type gives a valid distribution provided β is positive and 0 <= delta <= 1.
This logical function takes on a value of TRUE if the parameter values given produce a valid statistical distribution and FALSE if they don't
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.
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.
The generalized lambda functions GeneralisedLambdaDistribution
gl.check.lambda(c(0,1,.23,4.5),vect=TRUE) ## TRUE gl.check.lambda(c(0,-1,.23,4.5),vect=TRUE) ## FALSE gl.check.lambda(c(0,1,0.5,-0.5),param="rs",vect=TRUE) ## FALSE gl.check.lambda(c(0,2,1,3.4,1.2),param="fm5",vect=TRUE) ## FALSE
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.