Carry out the “starship” estimation method for the generalised lambda distribution
Calculates estimates for the FMKL parameterisation of the generalised lambda
distribution on the basis of data, using the starship method.
The starship method is built on the fact that the
generalised lambda distribution
is a transformation of the uniform distribution. This method finds the
parameters that transform the data closest to the uniform distribution.
This function uses a grid-based search to find a suitable starting point (using
starship.adaptivegrid) then uses optim to find
the parameters that do this.
starship(data, optim.method = "Nelder-Mead", initgrid = NULL, param="FMKL", optim.control=NULL)
data |
Data to be fitted, as a vector |
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optim.method |
Optimisation method for |
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initgrid |
Grid of values of lambda 3 and
lambda 4
to try, in If it is left as NULL, the default grid depends on the parameterisation.
For
( For
( For
and
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param |
choose parameterisation:
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optim.control |
List of options for the optimisation step. See
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The starship method is described in King \& MacGillivray, 1999 (see references). It is built on the fact that the generalised lambda distribution 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 is implemented in 2 stages in this function. First a grid search is
carried out, over a small number of possible parameter values
(see starship.adaptivegrid for details). Then the minimum from
this search is given as a starting point for an optimisation of the
Anderson-Darling value using optim, with method given by optim.method
See references for details on parameterisations.
Returns a list, with
lambda |
A vector of length 4, giving the estimated parameters, in order, lambda 1 - location parameter lambda 2 - scale parameter lambda 3 - first shape parameter lambda 4 - second shape parameter |
grid.results |
output from the grid search - see
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optim |
output from the optim search -
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Darren Wraith, Darren.Wraith@studentmail.newcastle.edu.au
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.
# data <- rgl(100,0,1,.2,.2) # vstarship(data,optim.method="Nelder-Mead",initgrid=list(lcvect=(0:4)/10, # ldvect=(0:4)/10))
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