fit.fkml {gld}

Estimate parameters of the FKML parameterisation of the generalised lambda distribution

Description

Estimates parameters of the FKML parameterisation of the Generalised \lambda Distribution. Five estimation methods are available; Numerical Maximum Likelihood, Maximum Product of Spacings, Titterington's Method, the Starship (also available in the starship function, which uses the same underlying code as this for the fkml parameterisation), Trimmed L-Moments, L-Moments, Distributional Least Absolutes, and Method of Moments.

Usage

fit.fkml(x, method = "ML", t1 = 0, t2 = 0, 
  l3.grid = c(-0.9, -0.5, -0.1, 0, 0.1, 0.2, 0.4, 0.8, 1, 1.5), 
  l4.grid = l3.grid, record.cpu.time = TRUE, optim.method = "Nelder-Mead",
  inverse.eps = .Machine$double.eps, optim.control=list(maxit=10000), 
  optim.penalty=1e20, return.data=FALSE)

Arguments

x

Data to be fitted, as a vector

method

A character string, to select the estimation method. One of: ML for numerical Maximum Likelihood, MPS or MSP for Maximum Spacings Product, TM for Titterington's Method, SM for Starship Method, TL for method of Trimmed L-moments, Lmom for method of L-moments, DLA for the method of Distributional Least Absolutes, or Mom for method of Moments.

t1

Number of observations to be trimmed from the left in the conceptual sample, t_1 (A non-negative integer, only used by TL-moment estimation, see details section)

t2

Number of observations to be trimmed from the right in the conceptual sample, t_2 (A non-negative integer, only used by TL-moment estimation, see details section). These two arguments are restricted by t_1+t_2 < n, where n is the sample size

l3.grid

A vector of values to form the grid of values of \lambda_3 used to find a starting point for the optimisation.

l4.grid

A vector of values to form the grid of values of \lambda_4 used to find a starting point for the optimisation.

record.cpu.time

Boolean — should the CPU time used in fitting be recorded in the fitted model object?

optim.method

Optimisation method, use any of the options available under method of optim.

inverse.eps

Accuracy of calculation for the numerical determination of F(x), defaults to .Machine$double.eps.

optim.control

List of options for the optimisation step. See optim for details.

optim.penalty

The penalty to be added to the objective function if parameter values are proposed outside the allowed region

return.data

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

Details

Maximum Likelihood Estimation of the generalised lambda distribution (gld) proceeds by calculating the density of the data for candidate values of the parameters. Because the gld is defined by its quantile function, the method first numerically obtains F(x) by inverting Q(u), then obtains the density for that observation.

Maximum Product of Spacings estimation (sometimes referred to as Maximum Spacing Estimation, or Maximum Spacings Product) finds the parameter values that maximise the product of the spacings (the difference between successive depths, F_\theta(x_{(i+1)})-F_\theta(x_{(i)}), where F_\theta(x) is the distribution function for the candidate values of the parameters). See Dean (2013) and Cheng & Amin (1981) for details.

Titterington (1985) remarked that MPS effectively added an “extra observation”; there are N data points in the original sample, but N + 1 spacings in the expression maximised in MPS. Instead of using spacings between transformed data points, so method TM uses spacings between transformed, adjacently-averaged, data points. The spacings are given by D_i=F_\theta(z_{(i)})-F_\theta(z_{(i-1)}), where \alpha_1 = z_0 < z_1 < \dots < z_n = \alpha_2 and z_i = (x_{(i)} +x_{(i+1)})/2 for i = 1,2, ... n-1 (where \alpha_1 and \alpha_2 are the lower and upper bounds on the support of the distribution). This reduces the number of spacings to n and achieves concordance with the original sample size. See Titterington (1985) and Dean (2013) for details.

The starship is built on the fact that the g\lambda d 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 these calculated depths closest (as measured by the Anderson-Darling statistic) to a uniform distribution. See King & MacGillivray (1999) for details.

TL-Moment estimation chooses the values of the parameters that minimise the difference between the sample Trimmed L-Moments of the data and the Trimmed L-Moments of the fitted distribution. TL-Moments are based on inflating the conceptual sample size used in the definition of L-Moments. The t1 and t2 arguments to the function define the extent of trimming of the conceptual sample. Thus, the default values of t1=0 and t2=0 reduce the TL-Moment method to L-Moment estimation. t1 and t2 give the number of observations to be trimmed (from the left and right respectively) from the conceptual sample of size n+t_1+t_2. These two arguments should be non-negative integers, and t_1+t_2 < n, where n is the sample size. See Elamir and Seheult (2003) for more on TL-Moments in general, Asquith, (2007) for TL-Moments of the RS parameterisation of the gld and Dean (2013) for more details on TL-Moment estimation of the gld.

The method of distributional least absolutes (DLA) minimises the sum of absolute deviations between the order statistics and their medians (based on the candidate parameters). See Dean (2013) for more information.

Moment estimation chooses the values of the parameters that minimise the (sum of the squared) difference between the first four sample moments of the data and the first four moments of the fitted distribution.

Value

fit.fkml returns an object of class "starship" (regardless of the estimation method used).

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, data and optim detailed below).

The value of fit.fkml is a list containing the following components:

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

optim

output from the optim search, optim for details

cpu

A vector showing the computing time used, returned if record.cpu.time is TRUE

data

The data, if return.data is TRUE

Author(s)

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

Ben Dean, University of Newcastle [e-mail]

References

Asquith, W. H. (2007), L-Moments and TL-Moments of the Generalized Lambda Distribution, Computational Statistics & Data Analysis, 51, 4484–4496.

Cheng, R.C.H. & Amin, N.A.K. (1981), Maximum Likelihood Estimation of Parameters in the Inverse Gaussian Distribution, with Unknown Origin, Technometrics, 23(3), 257–263. https://www.jstor.org/stable/1267789

Dean, B. (2013) Improved Estimation and Regression Techniques with the Generalised Lambda Distribution, PhD Thesis, University of Newcastle https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:13503

Elamir, E. A. H., and Seheult, A. H. (2003), Trimmed L-Moments, Computational Statistics & Data Analysis, 43, 299–314.

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.

Titterington, D. M. (1985), Comment on ‘Estimating Parameters in Continuous Univariate Distributions’, Journal of the Royal Statistical Society, Series B, 47, 115–116.

See Also

starship GeneralisedLambdaDistribution

Examples

example.data <- rgl(200,c(3,1,.4,-0.1),param="fkml")
example.fit <- fit.fkml(example.data,"MSP",return.data=TRUE)
print(example.fit)
summary(example.fit)
plot(example.fit,one.page=FALSE)