glCheckPars: Function to check the validity of parameters of the...

In GeneralizedLambda: The Generalized Lambda Distribution

Description

Checks the validity of parameters of the generalized lambda. The tests are simple for the FMKL, VSK and FM5 parameterisations, and much more complex for the RS parameterisation.

Usage

glCheckPars(param, version = "FMKL", lambda5 = NULL)
glCheckParsFMKL(param)
glCheckParsRS(param)
glCheckParsVSK(param)
glCheckParsAS(param)
glCheckParsFMKL5(param, lambda5)

Arguments

param	Vector of length 4. Each element of the vector defines lambda1 to lambda4, where lambdas from 1 to 4 define the location, scale and first and second shape parameters of the distribution. Note that the numbering of the lambda parameters for the fmkl parameterisation is different to that used by Freimer, Mudholkar, Kollia and Lin.
lambda5	a skewness parameter, in the fm5 parameterisation.
version	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

Details

See GeneralizedLambda 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 a 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

Value

glCheckPars returns TRUE if the parameter values given produce a valid statistical distribution and "error" if they don't according to its specified parameterisation. The default parameterisation is FMKL and without specifying parameterisation, the following functions can be used:

glCheckParsFMKL is used for FMKL parameterisation glCheckParsRS is used for RS parameterisation glCheckParsVSK is used for VSK parameterisation glCheckParsAS is used for AS parameterisation glCheckParsFMKL5 is used for F5 parameterisation

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], http://tolstoy.newcastle.edu.au/~rking/

Jeong Min Kim, [e-mail]

David Scott, [e-mail]

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.

Gilchrist, Warren G. (2000) Modelling with Quantile Functions, Chapman \& Hall/CRC, London.

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.

See Also

The generalized lambda functions GeneralizedLambdaDistribution

Examples

glCheckPars(param = c(0,1,.23,4.5)) ## TRUE
## Not run: 
  glCheckPars(c(0,-1,.23,4.5)) ## error
  glCheckPars(c(0,1,0.5,-0.5), param = "RS") ## error
  glCheckPars(c(0,2,1,3.4,1.2), param = "fm5") ## error

## End(Not run)
glCheckParsFMKL(param = c(1,2,0.5,-2)) ## TRUE
## Not run: 
  glCheckParsRS(param = c(-1,0.23,0.5,-2)) ## error
  glCheckParsVSK(param = c(0.1,-2,0,-2)) ## error

  glCheckParsAS(param = c(1,2,0,0)) ## Not yet implemented
  glCheckParsFMKL5(param = c(1,2,0.5,-2)) ## Not yet implemented

## End(Not run)

GeneralizedLambda documentation built on May 2, 2019, 4:50 p.m.