R: Carry out the "starship" estimation method for the...

starship {gld} | R Documentation

## Carry out the “starship” estimation method for the generalised lambda distribution

### Description

Estimates parameters 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 (`gld`) 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.

### Usage

    starship(data, optim.method = "Nelder-Mead", initgrid = NULL,
    inverse.eps = .Machine$double.eps, param="FMKL", optim.control=NULL, return.data=FALSE)

### Arguments

`data` | Data to be fitted, as a vector
`optim.method` | Optimisation method for `optim` to use, defaults to Nelder-Mead
`initgrid` | Grid of values of lambda 3 and lambda 4 to try, in `starship.adaptivegrid`. This should be a list with elements, `lcvect`, a vector of values for lambda 3, `ldvect`, a vector of values for lambda 4 and `levect`, a vector of values for lambda 5 (`levect` is only required if `param` is `fm5`). If it is left as NULL, the default grid depends on the parameterisation. For `fmkl`, both lcvect and ldvect default to:
-1.5 | -1 | -0.5 | -0.1 | 0 | 0.1 | 0.2 | 0.4 | 0.8 | 1 | 1.5
(`levect` is NULL). For `rs`, both lcvect and ldvect default to:
0.1 | 0.2 | 0.4 | 0.8 | 1 | 1.5
(`levect` is NULL). Note that this restricts the estimates to only part of the region of the lambda 3, lambda 4 plane. For `gpd`, the defaults are: δ:
0.3 | 0.5 | 0.7
and λ:
-1.5 | -.5 | 0 | .2 | .4 | 0.8 | 1.5 | 5
For `fm5`, both lcvect and ldvect default to:
-1.5 | -1 | -.5 | -0.1 | 0 | 0.1 | 0.2 | 0.4 | 0.8 | 1 | 1.5
and `levect` defaults to:
-0.5 | 0.25 | 0 | 0.25 | 0.5
`inverse.eps` | Accuracy of calculation for the numerical determination of F(x), defaults to `.Machine$double.eps`
`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)
`optim.control` | List of options for the optimisation step. See `optim` for details. If left as NULL, the parscale control is set to scale lambda 1 and lambda 2 by the absolute value of their starting points.
`return.data` | Logical: Should the function return the data (from the argument `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 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 `GeneralisedLambdaDistribution` for details on parameterisations.

### Value

`starship` returns an object of `class` `"starship"`.

`print` prints the estimated values of the parameters, while `summary.starship` prints these by default, but can also provide details of the estimation process (from the components `grid.results` and `optim` detailed below).

An object of class `"starship"` is a list containing at least the following components:

`lambda` | A vector of length 4 (or 5, for the fm5 parameterisation), giving the estimated parameters, in order, lambda 1 - location parameter lambda 2 - scale parameter lambda 3 - first shape parameter lambda 4 - second shape parameter (See `gld` for details of the parameters in the fm5 parameterisation) In the gpd parameterisation, the parameters are labelled: alpha - location parameter beta - scale parameter delta - skewness parameter lambda - tailweight parameter
`grid.results` | output from the grid search - see `starship.adaptivegrid` for details
`optim` | output from the optim search - `optim` for details

### Author(s)

Robert King, robert.king@newcastle.edu.au, https://tolstoy.newcastle.edu.au/~rking/

Darren Wraith

### 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.

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

### See Also

`starship.adaptivegrid`, `starship.obj`

### Examples

    data <- rgl(100,0,1,.2,.2)
    starship(data,optim.method="Nelder-Mead",initgrid=list(lcvect=(0:4)/10,
    ldvect=(0:4)/10))

[Package gld version 2.6.2 Index]