fit.gpd {gld}

Estimate parameters of the GPD type generalised lambda distribution

Description

Estimates parameters of the GPD type generalised \lambda Distribution. Estimation is via method of L moments or the starship method.

The Method of L-Moments estimates for the GPD type are the only estimates for any generalised lambda distribution type with closed form expressions, and the only with algebraic results for standard errors of the estimates.

For further details on the starship method, see starship.

Usage

fit.gpd(x, method = "LM", na.rm = TRUE, record.cpu.time = TRUE, return.data = FALSE,
    LambdaZeroEpsilon=1e-15)
fit.gpd.lmom(data, na.rm = TRUE,LambdaZeroEpsilon=1e-15)
fit.gpd.lmom.given(lmoms, n = NULL,LambdaZeroEpsilon=1e-15)

Arguments

x

Data to be fitted, as a vector

method

A character string, to select the estimation method. The available methods are 1. "LM" for Method of L-Moments, 2. "SM" or "starship" for the Starship method.

na.rm

Logical: Should missing values be removed?

record.cpu.time

Logical: should the CPU time used in fitting be recorded in the fitted model object?

return.data

Logical: Should the function return the data (from the argument x)?

data

Data to be fitted, as a vector

lmoms

A numeric vector containing two L-moments and two L-moment ratios, in the order l_1,l_2,t_3,t_4.

n

the sample size, defaults to NULL

LambdaZeroEpsilon

tolerance for lambda estimate of zero

Details

The starship method calls the starship function - see its help for more details.

The method of L-Moments equates sample L-Moments with expressions for the L-Moments of the GPD type GLD. Closed form expressions exist to give these estimates.

For many values there are two possible estimates for the same L Moment values, one in each of two regions of the GPD GLD parameter space, denoted region A and region B in van Staden (2013). More details on these regions can be found on page 154 of van Staden (2013).

If the 4th L-Moment ratio, \tau_4 is less than the minimum value that \tau_4 can obtain for the GPD generalised lambda distribution;

\tau_4^{(min)}=\frac{12-5\sqrt{6}}{12+5\sqrt{6}} \approx -0.0102051,

there is no possible L-Moment estimate (from either region A or B), and this function returns NA for the estimates.

When estimating from data, or for given L-Moments with n given, standard errors of the estimates are calculated if possible (standard errors are only finite if \lambda > -0.5).

If \lambda is zero, the GPD gld is a special case the Quantile Based Skew Logistic Distribution. If the estimated \lambda is within LambdaZeroEpsilon of zero, standard errors for alpha, beta and delta are calculated for the Quantile Based Skew Logistic Distribution and NA is returned as the standard error of \lambda.

Value

These functions return an object of class "GldGPDFit". It is a list, containing these components (optional components noted here);

estA

The estimate in region A. This will be NULL if there is no estimate in region A

estB

The estimate in region B. This will be NULL if there is no estimate in region B

warn

(only if estA and estB are both NULL), the reason there are no estimates. If this is the case, the function also issues a warning.

cpu

A vector showing the computing time used, returned if record.cpu.time is TRUE (only for fit.gpd).

data

The data, if return.data is TRUE (only for fit.gpd).

param

The character "gpd", indicating the GPD type of the generalised lambda distribution.

starship

The value returned by starship, if the starship method is used.

Each of the estimate elements (if they are not NULL) are either a vector of length 4, or a 4 by 2 matrix if standard errors are calculated. The elements of the vector, or rows of the matrix are the estimated parameters, in order;

alpha

location parameter

beta

scale parameter

delta

skewness parameter

lambda

kurtosis parameter

The columns of the matrix are the parameter, and its standard error.

Author(s)

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

Paul van Staden

References

Van Staden, Paul J., & M.T. Loots. (2009), Method of L-moment Estimation for the Generalized Lambda Distribution. In Proceedings of the Third Annual ASEARC Conference. Callaghan, NSW 2308 Australia: School of Mathematical and Physical Sciences, University of Newcastle.

See Also

GeneralisedLambdaDistribution

Examples

fit.gpd.lmom.given(c(1,.3,.6,.8))
example.data = rgl(n=300,c(5,2,0.8,-0.1),param="gpd")
fit.gpd(example.data)
fit.gpd(example.data,method="starship")