The L-moments page
An L-moment ratio diagram.
L-moments
L-moments are summary statistics for probability distributions
and data samples.
They are analogous to ordinary moments -- they provide measures of
location, dispersion, skewness, kurtosis, and other aspects of the shape
of probability distributions or data samples -- but are computed from
linear combinations of the ordered data values (hence the prefix L).
L-moments have the following theoretical advantages over ordinary
moments:
- For L-moments of a probability distribution to be meaningful, we
require only that the distribution have finite mean; no higher-order
moments need be finite [J. R. M. Hosking,
J. R. Statist. Soc. B, 52 (1990), Theorem 1].
- For standard errors of L-moments to be finite, we require only that
the distribution have finite variance; no higher-order moments need
be finite [Hosking, 1990, Theorem 3].
- Although moment ratios can be arbitrarily large, sample moment ratios
have algebraic bounds [J. Dalen, Statistics and Probability Letters,
5 (1987)]; sample L-moment ratios can take
any values that the corresponding population quantities can [Hosking,
1990, page 115].
In addition, the following properties hold in a wide range of
practical situations:
- Asymptotic approximations to sampling distributions are better for
L-moments than for ordinary moments [Hosking, 1990, Figure 4].
- L-moments are less sensitive to outlying data values
[P. Royston, Statistics in Medicine, 11 (1992),
Figure 7;
R. M. Vogel and N. M. Fennessey, Water Resources Research,
29 (1993), Figures 3 and 4].
- L-moments provide better identification of the parent distribution
that generated a particular data sample
[Hosking, 1990, Figure 6].
Click here to see a
formal definition of L-moments.
The L-moment ratio diagram can be used to compare
the L-skewness--L-kurtosis
relations of different distributions and data samples.
Click here
to see an example of the use of the L-moment ratio diagram.
Click here
to see approximations useful for drawing the curves
on L-moment ratio diagrams such as the one at the top of this page.
L-moments can be used as the basis of a unified approach
to the statistical analysis of univariate probability distributions.
They can be defined for any random variable whose mean exists
and form the basis of a general theory that covers:
- the summarization and description of theoretical probability distributions;
- the summarization and description of observed data samples;
- estimation of parameters and quantiles of probability distributions;
- hypothesis tests for probability distributions.
A short summary of the theory and applications of L-moments
can be found in the article "L-moments" in the
Encyclopedia of statistical sciences, Update Volume 2,
ed. S. Kotz, C. Read and D. L. Banks, Wiley, New York, 1998, pp. 357-362.
More details are given in the principal references,
the first two items in the following list of publications.
Publications related to L-moments
Software
- The LMOMENTS package contains Fortran routines
for L-moment computations and regional frequency analysis.
Version 3.03, containing 63 routines, is available from the
StatLib
software repository at Carnegie Mellon University.
Last modified: 12 December 2003.
[ J. R. M. Hosking's home page |
IBM Research home page ][
IBM home page |
Order |
Search |
Contact IBM |
Help |
(C) |
(TM)
]