R functions for use with the method of \(L\)-moments
Some functions support the trimmed \(L\)-moments defined by Elamir and Seheult (2003). Trimmed \(L\)-moments are based on linear combinations of order statistics that give zero weight to the most extreme order statistics and thereby can be defined for very heavy-tailed distributions that do not have a finite mean.
Function samlmu can compute sample trimmed \(L\)-moments.
Functions lmrp and lmrq can compute trimmed \(L\)-moments
of probability distributions.
Functions pelp and pelq can calculate parameters
of a probability distribution given its trimmed \(L\)-moments.
The distribution-specific functions lmr... and pel... and the
functions for \(L\)-moment ratio diagrams (lmrd, etc.) currently
do not support trimmed \(L\)-moments.
The functions cdf... (cumulative distribution functions)
and qua... (quantile functions) expect the distribution
parameters to be specified as a single vector.
This differs from the standard R convention, in which
each parameter is a separate argument.
There are two reasons for this.
First, the single-vector parametrization is consistent with
the Fortran routines on which these R functions are based.
Second, the single-vector parametrization is often easier to use.
For example, consider computing the 80th and 90th percentiles
of a normal distribution fitted to a set of \(L\)-moments
stored in a vector lmom.
In the single-vector parametrization, this is achieved by
quanor( c(.8,.9), pelnor(lmom) )
The separate-arguments parametrization would need a more complex expression, such as
do.call( qnorm, c( list(.8,.9), pelnor(lmom) ) )
In functions (lmrp, lmrq, pelp, pelq, evplot,
evdistp, evdistq) that take a cumulative distribution function
or a quantile function as an argument, the cumulative distribution function
or quantile function can use either form of parametrization.
Functions cdf..., qua..., lmr..., pel...,
and samlmu are analogous to Fortran routines from
the LMOMENTS package, version 3.04, available from StatLib at
https://lib.stat.cmu.edu/general/lmoments.
Functions cdfwak and samlmu, and all the lmr... and pel...
functions, internally call Fortran code that is derived from the
LMOMENTS package.
J. R. M. Hosking jrmhosking@gmail.com
\(L\)-moments are measures of the location, scale, and shape of probability distributions or data samples. They are based on linear combinations of order statistics. Hosking (1990) and Hosking and Wallis (1997, chap. 2) give expositions of the theory of \(L\)-moments and \(L\)-moment ratios. Hosking and Wallis (1997, Appendix) give, for many distributions in common use, expressions for the \(L\)-moments of the distributions and algorithms for estimating the parameters of the distributions by equating sample and population \(L\)-moments (the “method of \(L\)-moments”). This package contains R functions that should facilitate the use of \(L\)-moment-based methods.
For each of 13 probability distributions, the package contains functions to evaluate the cumulative distribution function and quantile function of the distribution, to calculate the \(L\)-moments given the parameters and to calculate the parameters given the low-order \(L\)-moments. These functions are as follows.
cdf...
computes the cumulative distribution function of the distribution.
qua...
computes the quantile function (inverse cumulative distribution function)
of the distribution.
lmr...
calculates the \(L\)-moment ratios of the distribution given its
parameters.
pel...
calculates the parameters of the distribution given its \(L\)-moments.
When the \(L\)-moments are the sample \(L\)-moments of a set of data,
the resulting parameters are of course the
“method of \(L\)-moments” estimates of the parameters.
Here ... is a three-letter code used to identify the distribution,
as given in the table below.
For example the cumulative distribution function of the gamma distribution
is cdfgam.
exp | exponential | |
gam | gamma | |
gev | generalized extreme-value | |
glo | generalized logistic | |
gpa | generalized Pareto | |
gno | generalized normal | |
gum | Gumbel (extreme-value type I) | |
kap | kappa | |
ln3 | lognormal | |
nor | normal | |
pe3 | Pearson type III | |
wak | Wakeby | |
wei | Weibull |
The following functions are also contained in the package.
samlmu
computes the sample \(L\)-moments of a data vector.
lmrp and lmrq
compute the \(L\)-moments of a probability distribution specified
by its cumulative distribution function (for function lmrp)
or its quantile function (for function lmrq).
The computation uses numerical integration applied to
a general expression for the \(L\)-moments of a distribution.
Functions lmrp and lmrq can be used for any univariate
distribution. They are slower and usually less accurate than the
computations carried out for specific distributions by the
lmr... functions.
pelp and pelq
compute the parameters of a probability distribution
as a function of the \(L\)-moments.
The computation uses function lmrp or lmrq to compute
\(L\)-moments and numerical optimization to find parameter values
for which the sample and population \(L\)-moments are equal.
Functions pelp and pelq can be used for any univariate
distribution. They are slower and usually less accurate than the
computations carried out for specific distributions by the
pel... functions.
lmrd
draws an \(L\)-moment ratio diagram.
lmrdpoints and lmrdlines
add points, or connected line segments, respectively,
to an \(L\)-moment ratio diagram.
evplot
draws an “extreme-value plot”, i.e. a quantile-quantile plot
in which the horizontal axis is the quantile of an
extreme-value type I (Gumbel) distribution.
evpoints, evdistp, and evdistq
add, respectively, a set of points, a cumulative distribution function,
and a quantile function to an extreme-value plot.
Elamir, E. A. H., and Seheult, A. H. (2003). Trimmed L-moments. Computational Statistics and Data Analysis, 43, 299-314.
Hosking, J. R. M. (1990). \(L\)-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Series B, 52, 105-124.
Hosking, J. R. M., and Wallis, J. R. (1997). Regional frequency analysis: an approach based on L-moments. Cambridge University Press.