# lmom-package: The lmom package In lmom: L-Moments

 lmom-package R Documentation

## 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`.

 `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.

### Trimmed `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.

### 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 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.

### 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 jrmhosking@gmail.com

### References

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.

lmom documentation built on Aug. 29, 2023, 9:07 a.m.