starship function, which uses the same underlying code as this for the fkml parameterisation),
and Trimmed L-Moments.
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$double.eps, optim.control=list(maxit=10000), optim.penalty=1e20, return.data=FALSE)ML for Numerical Maximum Likelihood, MPS or MSP
for Maximum Product of Spacings, TM for Titterington's Method, SM for Starship Method, TL for Method of TL-moments, or DLA for the method of distributional least absolutes.method of optim..Machine$double.eps.optim for details.
data)?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: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, $F(x_(i+1);theta)-F(x_(i);theta)$, where $F(x;theta)$ 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 The starship is built on the fact that the $gld$
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 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 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.
TM
uses spacings between transformed, adjacently-averaged, data points.
The spacings are given by $Di=F(z(i);theta)- F(z(i-1);theta)$,
where $alpha1 = z0 < z1 < ...
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 $n+t1+t2$. These two
arguments should be non-negative integers, and $t1+t2 < n$, 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.
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. http://www.jstor.org/stable/1267789
Dean, B. (2013) Improved Estimation and Regression Techniques with the Generalised Lambda Distribution, PhD Thesis, University of Newcastle http://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.
starship
GeneralisedLambdaDistribution
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)
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