The lmom package

Description

R functions for use with the method of L-moments

Details

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.

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.

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.

Parameters of cumulative distribution functions and quantile functions

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 parameterization 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) )

  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.

Relation to the LMOMENTS Fortran package

Functions cdf..., qua..., lmr..., pel..., and samlmu are analogous to Fortran routines from the LMOMENTS package, version 3.04, available from StatLib at http://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.

Author(s)

J. R. M. Hosking hosking@watson.ibm.com

References

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.