gld (version 2.4.1)

starship.obj: Objective function that is minimised in starship estimation method

Description

The starship is a method for fitting the generalised lambda distribution. See starship for more details.

This function is the objective funciton minimised in the methods. It is a goodness of fit measure carried out on the depths of the data.

Usage

starship.obj(par, data, inverse.eps, param = "fmkl")

Arguments

par

parameters of the generalised lambda distribution, a vector of length 4, giving $lambda 1$ to $lambda 4$. See GeneralisedLambdaDistribution for details on the definitions of these parameters

data

Data --- a vector

inverse.eps

Accuracy of calculation for the numerical determination of $F(x)$, defaults to $1e-8$

param

choose parameterisation: fmkl uses Freimer, Mudholkar, Kollia and Lin (1988) (default). rs uses Ramberg and Schmeiser (1974)

Value

The Anderson-Darling goodness of fit measure, computed on the transformed data, compared to a uniform distribution. Note that this is NOT the goodness-of-fit measure of the generalised lambda distribution with the given parameter values to the data.

Details

The starship method is described in King \& MacGillivray, 1999 (see references). It is built on the fact that the generalised lambda distribution (gld) is a transformation of the uniform distribution. Thus the inverse of this transformation is the distribution function for the gld. The starship method applies different values of the parameters of the distribution to the distribution function, calculates the depths q corresponding to the data and chooses the parameters that make the depths closest to a uniform distribution.

The closeness to the uniform is assessed by calculating the Anderson-Darling goodness-of-fit test on the transformed data against the uniform, for a sample of size length(data).

This function returns that objective function. It is provided as a seperate function to allow users to carry out minimisations using optim or other methods. The recommended method is to use the starship function.

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.

Ramberg, J. S. & Schmeiser, B. W. (1974), An approximate method for generating asymmetric random variables, Communications of the ACM 17, 78--82. King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353--374

Owen, D. B. (1988), The starship, Communications in Statistics - Computation and Simulation 17, 315--323.

http://tolstoy.newcastle.edu.au/~rking/gld/

See Also

starship, starship.adaptivegrid

Examples

data <- rgl(100,0,1,.2,.2)
starship.obj(c(0,1,.2,.2),data,inverse.eps=1e-10,"fmkl")