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)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
This can be either a single numeric value or a vector.
If it is a vector, it must be of length 4 for parameterisations
fmkl or rs and of length 5 for parameterisation fm5.
If it is a vector, it gives all the parameters of the generalised lambda
distribution (see below for details) and the other lambda arguments
must be left as NULL.
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.
\(\lambda_2\) - scale parameter (\(\beta\) for gpd)
\(\lambda_3\) - first shape parameter (\(\delta\), skewness parameter for gpd)
\(\lambda_4\) - second shape parameter (\(\lambda\), kurtosis parameter for gpd)
\(\lambda_5\) - a skewing parameter, in the fm5 parameterisation
choose parameterisation:
fmkl uses Freimer, Mudholkar, Kollia and Lin (1988) (default).
rs uses Ramberg and Schmeiser (1974)
fm5 uses the 5 parameter version of the FMKL parameterisation
(paper to appear)
A logical, set this to TRUE if the parameters are given in the
vector form (it turns off checking of the format of lambdas and the
other lambda arguments
Robert King, robert.king.newcastle@gmail.com, https://github.com/newystats/
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 \(-1 \leq \lambda_5 \leq 1\).
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 | \(>0\) | \(\ge 0\) | \(\ge 0\) | one of \(\lambda_3\) and \(\lambda_4\) must be non-zero |
| 4 | all | \(<0\) | \(\le 0\) | \(\le 0\) | 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
$$ \frac{(1-\lambda_3) ^{1-\lambda_3}(\lambda_4-1)^{\lambda_4-1}} {(\lambda_4-\lambda_3)^{\lambda_4-\lambda_3}} < - \frac{\lambda_3}{\lambda_4} $$
Equation 2
$$ \frac{(1-\lambda_4) ^{1-\lambda_4}(\lambda_3-1)^{\lambda_3-1}} {(\lambda_3-\lambda_4)^{\lambda_3-\lambda_4}} < - \frac{\lambda_4}{\lambda_3} $$
The GPD type gives a valid distribution provided \(\beta\) is positive and \(0 \leq \delta \leq 1\).
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
Run the code above in your browser using DataLab