gl.check.lambda(lambdas, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, param = "fkml", lambda5 = NULL, vect = FALSE)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.
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)lambdas and the
other lambda argumentsGeneralisedLambdaDistribution 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 | $<0$ <="" td=""> | $< -1$ |
| $> 1$ | 2 | all | $<0$ <="" td=""> | |
| $> 1$ | $< -1$ | 3 | all | |
| $>0$ | $\ge 0$ | $\ge 0$ | one of $lambda 3$ and $lambda 4$ must be non-zero | 4 |
| all | $<0$ <="" td=""> | $\le 0$ | $\le 0$ | one of $lambda 3$ and $lambda 4$ must be non-zero |
| 5 | all | $<0$ <="" td=""> | $> -1$ and $< 0$ | $>1$ |
| equation 1 below must also be satisfied | 6 | all | $<0$ <="" td=""> | $>1$ |
| $> -1$ and $< 0$ | equation 2 below must also be satisfied | region | $lambda 1$ | $lambda 2$ |
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
$$ \frac{(1-\lambda_4) ^{1-\lambda_4}(\lambda_3-1)^{\lambda_3-1}} {(\lambda_3-\lambda_4)^{\lambda_3-\lambda_4}} < - \frac{\lambda_4}{\lambda_3} $$
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.
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
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