gl.check.lambda {gld}

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

Description

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.

Usage

gl.check.lambda(lambdas, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, param = "fkml", 
  lambda5 = NULL, vect = FALSE)

Arguments

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 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.

lambda2

\lambda_2 - scale parameter (\beta for gpd)

lambda3

\lambda_3 - first shape parameter (\delta, skewness parameter for gpd)

lambda4

\lambda_4 - second shape parameter (\lambda, kurtosis parameter for gpd)

lambda5

\lambda_5 - a skewing parameter, in the fm5 parameterisation

param

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)

vect

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

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.

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.

Value

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

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, [e-mail], https://github.com/newystats/

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.

https://github.com/newystats/gld/

See Also

The generalized lambda functions GeneralisedLambdaDistribution

Examples

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