L-moments of a general probability distribution
Computes the L-moments or trimmed L-moments
of a probability distribution
given its cumulative distribution function (for function lmrp)
or quantile function (for function lmrq).
lmrp(pfunc, ..., bounds=c(-Inf,Inf), symm=FALSE, order=1:4,
ratios=TRUE, trim=0, acc=1e-6, subdiv=100, verbose=FALSE)
lmrq(qfunc, ..., symm=FALSE, order=1:4, ratios=TRUE, trim=0,
acc=1e-6, subdiv=100, verbose=FALSE)pfunc |
Cumulative distribution function. |
qfunc |
Quantile function. |
... |
Arguments to |
bounds |
Either a vector of length 2, containing the lower and upper bounds of the distribution, or a function that calculates these bounds given the distribution parameters as inputs. |
symm |
For For If the distribution is symmetric, odd-order L-moments are exactly zero and the symmetry is used to slightly speed up the computation of even-order L-moments. |
order |
Orders of the L-moments and L-moment ratios to be computed. |
ratios |
Logical. If |
trim |
Degree of trimming. If a single value, symmetric trimming of the specified degree will be used. If a vector of length 2, the two values indicate the degrees of trimming at the lower and upper ends of the “conceptual sample” (Elamir and Seheult, 2003) of order statistics that is used to define the trimmed L-moments. |
acc |
Requested accuracy. The function will try to achieve this level of accuracy, as relative error for L-moments and absolute error for L-moment ratios. |
subdiv |
Maximum number of subintervals used in numerical integration. |
verbose |
Logical. If |
Computations use expressions in Hosking (2007):
eq. (7) for lmrp, eq. (5) for lmrq.
Integrals in those expressions are computed by numerical integration.
If verbose=FALSE and ratios=FALSE,
a numeric vector containing the L-moments.
If verbose=FALSE and ratios=TRUE,
a numeric vector containing the L-moments (of orders 1 and 2)
and L-moment ratios (of orders 3 and higher).
If verbose=TRUE, a data frame with columns as follows:
value |
L-moments (if |
abs.error |
Estimate of the absolute error in the computed value. |
message |
|
pfunc and qfunc can be either the standard R form of
cumulative distribution function or quantile function
(i.e. for a distribution with r parameters, the first argument is the
variate x or the probability p and the next r arguments
are the parameters of the distribution) or the cdf... or
qua... forms used throughout the lmom package
(i.e. the first argument is the variate x or probability p
and the second argument is a vector containing the parameter values).
Even for the R form, however, starting values for the parameters
are supplied as a vector start.
If bounds is a function, its arguments must match
the distribution parameter arguments of pfunc:
either a single vector, or a separate argument for each parameter.
Arguments bounds, symm, order,
ratios, trim, acc, subdiv, and verbose
cannot be abbreviated and must be specified by their full names
(if abbreviated, the names would be matched to the arguments of
pfunc or qfunc).
In package lmom versions 1.6 and earlier, the “Details” section stated that
“Integrals in those expressions are computed by numerical integration,
using the R function integrate”.
As of version 2.0, numerical integration uses an internal function that directly calls
(slightly modified versions of) Fortran routines in QUADPACK (Piessens et al. 1983).
R's own integrate function uses C code “based on” the QUADPACK routines,
but in R versions 2.12.0 through 3.0.1 did not in every case reproduce the results
that would have been obtained with the Fortran code (this is R bug PR#15219).
J. R. M. Hosking jrmhosking@gmail.com
Elamir, E. A. H., and Seheult, A. H. (2003). Trimmed L-moments. Computational Statistics and Data Analysis, 43, 299-314.
Hosking, J. R. M. (2007). Some theory and practical uses of trimmed L-moments. Journal of Statistical Planning and Inference, 137, 3024-3039.
Piessens, R., deDoncker-Kapenga, E., Uberhuber, C., and Kahaner, D. (1983). Quadpack: a Subroutine Package for Automatic Integration. Springer Verlag.
lmrexp to compute (untrimmed) L-moments of specific distributions.
samlmu to compute (trimmed or untrimmed) L-moments of a data sample.
## Generalized extreme-value (GEV) distribution
## - three ways to get its L-moments
lmrp(cdfgev, c(2,3,-0.2))
lmrq(quagev, c(2,3,-0.2))
lmrgev(c(2,3,-0.2), nmom=4)
## GEV bounds specified as a vector
lmrp(cdfgev, c(2,3,-0.2), bounds=c(-13,Inf))
## GEV bounds specified as a function -- single vector of parameters
gevbounds <- function(para) {
k <- para[3]
b <- para[1]+para[2]/k
c(ifelse(k<0, b, -Inf), ifelse(k>0, b, Inf))
}
lmrp(cdfgev, c(2,3,-0.2), bounds=gevbounds)
## GEV bounds specified as a function -- separate parameters
pgev <- function(x, xi, alpha, k)
pmin(1, pmax(0, exp(-((1-k*(x-xi)/alpha)^(1/k)))))
pgevbounds <- function(xi,alpha,k) {
b <- xi+alpha/k
c(ifelse(k<0, b, -Inf), ifelse(k>0, b, Inf))
}
lmrp(pgev, xi=2, alpha=3, k=-0.2, bounds=pgevbounds)
## Normal distribution
lmrp(pnorm)
lmrp(pnorm, symm=0)
lmrp(pnorm, mean=2, sd=3, symm=2)
# For comparison, the exact values
lmrnor(c(2,3), nmom=4)
# Many L-moment ratios of the exponential distribution
# This may warn that "the integral is probably divergent"
lmrq(qexp, order=3:20)
# ... nonetheless the computed values seem accurate:
# compare with the exact values, tau_r = 2/(r*(r-1)):
cbind(exact=2/(3:20)/(2:19), lmrq(qexp, order=3:20, verbose=TRUE))
# Of course, sometimes the integral really is divergent
## Not run:
lmrq(function(p) (1-p)^(-1.5))
## End(Not run)
# And sometimes the integral is divergent but that's not what
# the warning says (at least on the author's system)
lmrp(pcauchy)
# Trimmed L-moments for Cauchy distribution are finite
lmrp(pcauchy, symm=0, trim=1)
# Works for discrete distributions too, but often requires
# a larger-than-default value of 'subdiv'
lmrp(ppois, lambda=5, subdiv=1000)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.