The moments by ilicaifengba


									The L-moments page by JRM Hosking

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].
An L-moment ratio diagram.

L-moments for data samples
Probability weighted moments, defined by J. A. Greenwood et al. [Water Resources Research,
15 (1979), 1049-1054], are precursors of L-moments. Sample probability weighted moments,
computed from data values X1, X2, ... Xn, arranged in increasing order, are given by

L-moments are certain linear combinations of probability weighted moments that have simple
interpretations as measures of the location, dispersion and shape of the data sample. The first few
L-moments are defined by
(the coefficients are those of the "shifted Legendre polynomials").
The first L-moment is the sample mean, a measure of location. The second L-moment is (a
multiple of) Gini's mean difference statistic, a measure of the dispersion of the data values about
their mean.
By dividing the higher-order L-moments by the dispersion measure, we obtain the L-moment

These are dimensionless quantities, independent of the units of measurement of the data. t3 is a
measure of skewness and t4 is a measure of kurtosis -- these are respectively the L-skewness and
L-kurtosis. They take values between -1 and +1 (exception: some even-order L-moment ratios
computed from very small samples can be less than -1).
The L-moment analogue of the coefficient of variation (standard deviation divided by the mean),
is the L-CV, defined by

It takes values between 0 and 1.

L-moments for probability distributions
For a probability distribution with cumulative distribution function F(x), probability weighted
moments are defined by

L-moments are defined in terms of probability weighted moments, analogously to the sample L-

L-moment ratios are defined by


      Uniform (rectangular) distribution on (0,1):
      Normal distribution with mean 0 and variance 1:

exponential distribution with mean 1:

The L-moment ratio diagram can be used to compare the L-skewness--L-kurtosis relations of
different distributions and data samples.

An example of the use of the L-moment ratio diagram
The L-moment ratio diagram can be used to compare the L-skewness--L-kurtosis relations of
different distributions and data samples. This gives a visual indication of which distribution may
be expected to give a good fit to a data sample or samples. The following example is adapted
from J. R. M. Hosking ["L-moments: analysis and estimation of distributions using linear
combinations of order statistics", Journal of the Royal Statistical Society, Series B, 52 (1990),
105-124, section 3.5], and appears here by permission of the Royal Statistical Society.
Annual maximum hourly rainfall data at 689 raingages in California was analysed by the State of
California ["Rainfall depth-duration-frequency for California", Technical Report, State of
California Department of Water Resources, 1981]. It was suggested that a gamma distribution
was appropriate for the data because the average values of sample skewness and kurtosis were
consistent with the relationship between the population skewness and kurtosis of gamma
distributions. This inference is valid only if sample moments are accurate estimators of
population moments. For the California data, this is dubious, because the sample sizes are small -
- only 12 gauges have records as long as 50 years. L-moments tell a different story.
The sample L-skewness and L-kurtosis values for the 68 sites with at least 20 years of record in
the Central Valley of California are shown in the first diagram. The data are on average closer to
the population L-moments of a generalized extreme-value (GEV) distribution rather than a
gamma distribution.

The second diagram shows sample L-moments of data simulated from independent GEV
distributions each with population L-skewness 0.24 (the average of the 68 sample L-skewness
values) and the same record lengths as the actual rainfall records. The scatter of the points is
similar to the first diagram. This is what one would expect if the actual data did in fact follow a
GEV distribution.

Sample L-moments of data simulated from a gamma distribution are shown in the third diagram.
This has an appearance rather different from the first two diagrams: there are many fewer points
above the "GEV" line and many more below the "gamma" line.
We therefore conclude that the Central Valley hourly rainfall data may be well described by a
GEV distribution but seem most unlikely to follow a gamma distribution.

Approximations for use in constructing L-moment ratio

To construct an L-moment ratio diagram it is convenient to have simple explicit expressions for
tau_4 (L-kurtosis) in terms of tau_3 (L-skewness) for some commonly used probability
distribtions. Polynomial approximations of the form
                                               2                    8
 tau_4    = A0    + A1 * (tau_3) + A2 * (tau_3) + ... + A8 * (tau_3)
have been obtained, and the coefficients are given in the table below. "Overall lower bound" is
the lower bound on tau_4 for all distributions [Hosking, J. R. Statist. Soc. B, 1990, eq.(2.7)]. For
given tau_3, the approximations yield values of tau_4 that are accurate to within 0.0005
provided that tau_3 is in the range -0.9 to +0.9, except that for the generalized extreme-value
distribution 0.0005 accuracy is attained only when tau_3 is between -0.6 and +0.9,

The approximations are not intended for detailed analytical calculations -- for that purpose, use
the routines in the LMOMENTS software package -- but they are sufficiently accurate for use in
plotting theoretical L-moment relationships on an L-moment ratio diagram.
       Generalized Generalized    Generalized             Pearson    Overall
        logistic   extreme-value    Pareto     Lognormal type III lower
 A0      0.16667      0.10701       0.          0.12282   0.12240    -0.25
 A1       .           0.11090       0.20196      .         .           .
 A2      0.83333      0.84838       0.95924     0.77518   0.30115     1.25
 A3       .          -0.06669      -0.20096      .         .           .
 A4       .           0.00567       0.04061     0.12279   0.95812      .
 A5       .          -0.04208        .           .         .           .
 A6       .           0.03673        .         -0.13638 -0.57488       .
 A7       .            .             .           .         .           .
 A8       .            .             .          0.11368   0.19383      .
This material is based on an IBM Research Report by J. R. M. Hosking ["Approximations for
use in constructing L-moment ratio diagrams", Research Report RC 16635, IBM Research
Division, Yorktown Heights, N.Y., 1991]. The approximations are also given in Appendix A.12
of the book Regional frequency analysis: an approach based on L-moments, by J. R. M. Hosking
and J. R. Wallis.
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.

Book publication related to L-moments
Hosking, J. R. M., and Wallis, J. R. (1997). Regional frequency analysis: an approach based on
L-moments. Cambridge University Press, Cambridge, U.K.
Extreme environmental events, such as floods, droughts, rainstorms, and high winds, have severe
consequences for human society. How frequently an event of a given magnitude may be
expected to occur is of great importance. Planning for weather-related emergencies, design of
civil engineering structures, reservoir management, pollution control, and insurance risk
calculations, all rely on knowledge of the frequency of these extreme events. Estimation of these
frequencies is difficult because extreme events are by definition rare and data records are often
Regional frequency analysis resolves this problem by 'trading space for time': data from several
sites are used in estimating event frequencies at any one site. L-moments are a recent
development within statistics. They form the basis of an elegant mathematical theory in their
own right, and can be used to facilitate the estimation process in regional frequency analysis. L-
moment methods are demonstrably superior to those that have been used previously, and are now
being adopted by major organizations worldwide.
This book is the first complete account of the L-moment approach to regional frequency analysis.
It brings together results that previously were scattered among academic journals, and also
includes much new material. Regional frequency analysis comprehensively describes the
theoretical background to the subject, is rich in practical advice for users, and contains detailed
examples that illustrate the approach. This book will be of great value to hydrologists,
atmospheric scientists and civil engineers, concerned with environmental extremes. ISBN 0-521-
43045-3. Published May 1997.

Papers describing L-moments and their properties
Chen, G., and Balakrishnan, N. (1995). The infeasibility of probability weighted moments
estimation of some generalized distributions. In Recent advances in life-testing and reliability,
ed. N. Balakrishnan, 565-573. Boca Raton, Fla.: CRC Press.

Donaldson, R. W. (1996). Calculating inverse Cv, skew and PWM functions for Pearson-3, log-
normal, extreme-value, and log-logistic distributions. Communications in Statistics - Simulation
and Computation, 25, 741-747.

Elamir, E. A. H., and Seheult, A. H. (2003). Trimmed L-moments. Computational Statistics and
Data Analysis, 43, 299-314.

Gingras, D., and Adamowski, K. (1994). Performance of L-moments and nonparametric flood
frequency analysis. Canadian Journal of Civil Engineering, 21, 856-862.

Guttman, N. B. (1994). On the sensitivity of sample L-moments to sample size. Journal of
Climate, 7, 1026-1029.

Haktanir, T., and Bozduman, A. (1995). A study on sensitivity of the probability-weighted
moments method on the choice of the plotting position formula. Journal of Hydrology, 168, 265-
Lu, L.-H., and Stedinger, J. R. (1992). Sampling variance of normalized GEV/PWM quantile
estimators and a regional homogeneity test. Journal of Hydrology, 138, 223-245.

Lu, L.-H., and Stedinger, J. R. (1992). Variance of two- and three-parameter GEV/PWM
quantile estimators: formulae, confidence intervals, and a comparison. Journal of Hydrology,
138, 247-267.

Pandey, M. D., van Gelder, P. H. A. J. M., and Vrijling, J. K. (2001). Assessment of an L-
kurtosis-based criterion for quantile estimation. Journal of Hydrologic Engineering, 6, 284-292.

Peel, M. C., Wang, Q. J., Vogel, R. M., and McMahon, T. A. (2001). The utility of L-moment
ratio diagrams for selecting a regional probability distibution. Hydrological Sciences Journal, 46,

Royston, P. (1992). Which measures of skewness and kurtosis are best? Statistics in Medicine,
11, 333-343.

Sankarasubramanian, A., and Srinivasan, K. (1999). Investigation and comparison of sampling
properties of L-moments and conventional moments. Journal of Hydrology, 218, 13-34.

Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E. (1992). Frequency analysis of extreme
events. In Handbook of Hydrology, ed. D. R. Maidment, chapter 18. New York: McGraw-Hill.

Ulrych, T. J., Velis, D. R., Woodbury, A. D., and Sacchi, M .D. (2000). L-moments and C-
moments. Stochastic Environmental Research and Risk Assessment, 14, 50-68.

Vogel, R. M., and Fennessey, N. M. (1993). L-moment diagrams should replace product-moment
diagrams. Water Resources Research, 29, 1745-1752.

Wallis, J. R. (1988). Catastrophes, computing, and containment: living with our restless habitat.
Speculations in Science and Technology, 11, 295-324.

Wang, Q. J. (1996). Direct sample estimators of L moments. Water Resources Research, 32,

Wang, Q. J. (1997). LH moments for statistical analysis of extreme events. Water Resources
Research, 33, 2841-2848.

Zafirakou-Koulouris, A., Habermeier, J., Craig, S. M., and Vogel, R. M. (1998). L-moment
diagrams for censored observations. Water Resources Research, 34, 1241-1249..

Papers using L-moments in the environmental sciences
Adamowski, K. (2000). Regional analysis of annual maximum and partial duration flood data by
nonparametric and L-moment methods. Journal of Hydrology, 229, 219-231.

Anctil, F., Larouche, W., and Hoang, V. D. (2000). Analyse regionale des etiages 7-jours de la
province de Quebec. Water Quality Research Journal of Canada, 35, 125-146.

Awadallah, A. G., Rousselle, J., and Leconte, R. (1999). Evolution du risque hydrologique sur la
riviere Chateauguay. Canadian Journal of Civil Engineering, 26, 510-523.

Benson, C. (1993). Probability distributions for hydraulic conductivity of compacted soil liners.
Journal of Geotechnical Engineering, 119, 471-486.

Ben-Zvi, A., and Azmon, B. (1997). Joint use of L-moment diagram and goodness-of-fit test: a
case study of diverse series. Journal of Hydrology, 198, 245-259.

Bradley, A. A. (1998). Regional frequency analysis methods for evaluating changes in
hydrologic extremes. Water Resources Research, 34, 741-750.

Burn, D. H. (2002). The use of resampling for estimating confidence intervals for single site and
pooled frequency analysis. Hydrological Sciences Journal, 48, 25-38.

Castellarin, A., Burn, D. H., and Brath, A. (2001). Assessing the effectiveness of hydrological
similarity measures for flood frequency analysis. Journal of Hydrology, 241, 270-285.

Clausen, B., and Pearson, C. P. (1995). Regional frequency analysis of annual maximum
streamflow drought. Journal of Hydrology, 173, 111-130.

Cong, S., Li, Y., Vogel, J. L., and Schaake, J. C. (1993). Identification of the underlying
distribution form of precipitation by using regional data. Water Resources Research, 29, 1103-

Cunderlik, J. M., and Burn, D. H. (2003). Non-stationary pooled flood frequency analysis.
Journal of Hydrology, 276, 210-223.

Daviau, J.-L., Adamowski, K., and Patry, G. G. (2000). Regional flood frequency using GIS, L-
moment and geostatistical methods. Hydrological Processes, 14, 2731-2753.

Dewar, R. E., and Wallis, J. R. (1999). Geographical patterning of interannual rainfall variability
in the tropics and near tropics: an L-moments approach. Journal of Climate, 12, 3457-3466.

Duan, J., Selker, J., and Grant, G. E. (1998). Evaluation of probability density functions in
precipitation models for the Pacific northwest. Journal of the American Water Resources
Association, 34, 617-627.

Fill, H. D., and Stedinger, J. R. (1995). L moment and PPCC goodness-of-fit tests for the
Gumbel distribution and impact of autocorrelation. Water Resources Research, 31, 225-229.

Fill, H. D., and Stedinger, J. R. (1999). Using regional regression procedures within index flood
procedures and an empirical Bayesian estimator. Journal of Hydrology, 210, 128-145.
Gellens, D. (2003). Combining regional approach and data extension procedure for assessing
GEV distribution of extreme precipitation in Belgium. Journal of Hydrology, 268, 113-126.

Gingras, D., Adamowski, K., and Pilon, P. J. (1994). Regional flood equations for the provinces
of Ontario and Quebec. Water Resources Bulletin, 30, 55-67.

Glaves, R., and Waylen, P. R. (1997). Regional flood frequency analysis in southern Ontario
using L-moments. The Canadian Geographer, 41, 178-193.

Gottschalk, L., Tallaksen, L. M., and Perzyna, G. (1997). Derivation of low flow distribution
functions using recession curves. Journal of Hydrology, 194, 239-262.

Guttman, N. B. (1993). The use of L-moments in the determination of regional precipitation
climates. Journal of Climate, 6, 2309-2325.

Guttman, N. B. (1996). Statistical characteristics of US historical climatology network
temperature distributions. Climate Research, 6, 33-43.

Heth, C. D., and Cornell, E. H. (1998). Characteristics of travel by persons lost in Albertan
wilderness areas. Journal of Environmental Psychology, 18, 223-235.

Kachroo, R. K., Mkhandi, S. H., and Parida, B. P. (2000). Flood frequency analysis of southern
Africa: I. Delineation of homogeneous regions. Hydrological Sciences Journal, 45, 437-447.

Kjeldsen, T. R., and Rosbjerg, D. (2002). Comparison of regional index flood estimation
procedures based on the extreme value type I distribution. Stochastic Environmental Research
and Risk Assessment, 16, 358-373.

Kjeldsen, T. R., Smithers, J. C., and Schulze, R. E. (2002). Regional flood frequency analysis in
the KwaZulu-Natal province, South Africa, using the index-flood method. Journal of Hydrology,
255, 194-211.

Kroll, C. N., and Vogel, R. M. (2002). Probability distribution of low streamflow series in the
United States. Journal of Hydrologic Engineering, 7, 137-146.

Kumar, P., Guttorp, P., and Foufoula-Georgiou, E. (1994). A probability-weighted moment test
to assess simple scaling. Stochastic Hydrology and Hydraulics, 8, 173-183.

Kumar, R., Chatterjee, C., and Kumar, S. (2003). Regional flood formulas using L-moments for
small watersheds of Sone subzone of India. Applied Engineering in Agriculture, 19, 47-53.

Lee, S. H., and Maeng, S. J. (2003a). Frequency analysis of extreme rainfall using L-moments.
Irrigation and Drainage, 52, 219-230.

Lee, S. H., and Maeng, S. J. (2003b). Comparison and analysis of design floods by the change in
the order of LH-moment methods. Irrigation and Drainage, 52, 231-245.
Lim, Y. H., and Lye, L. M. (2003). Regional flood estimation for ungauged basins in Sarawak,
Malaysia. Hydrological Sciences Journal, 48, 79-94.

Madsen, H., Pearson, C. P., and Rosbjerg, D. (1997). Comparison of annual maximum series and
partial duration series methods for modeling extreme hydrologic events. 2. Regional modeling.
Water Resources Research, 33, 759-769.

Madsen, H., and Rosbjerg, D. (1997). Generalized least squares and empirical Bayes estimation
in regional partial duration series index-flood modeling. Water Resources Research, 33, 771-

Madsen, H., Mikkelsen, P. S., Rosbjerg, D., and Harremoes, P. (1998). Estimation of regional
intensity-duration-frequency curves for extreme precipitation. Water Science and Technology,
37, 29-36.

Pandey, M. D., Van Gelder, P. H. A. J. M., and Vrijling, J. K. (2001). The estimation of extreme
quantiles of wind velocity using L-moments in the peaks-over-threshold approach. Structural
Safety, 23, 179-192.

Pandey, M. D., Van Gelder, P. H. A. J. M., and Vrijling, J. K. (2003). Bootstrap simulations for
evaluating the uncertainty associated with peaks-over-threshold estimates of extreme wind
velocity. Environmetrics, 14, 27-43.

Parida, B. (1999). Modelling of Indian summer monsoon rainfall using a four-parameter kappa
distribution. International Journal of Climatology, 19, 1389-1398.

Park, J.-S., Jung, H.-S., Kim, R.-S., and Oh, J.-H. (2001). Modelling summer extreme rainfall
over the Korean Peninsula using Wakeby distribution. International Journal of Climatology, 21,

Pearson, C. P. (1991). New Zealand regional flood frequency analysis using L moments. Journal
of Hydrology (NZ), 30, 53-64.

Pearson, C. P. (1993). Application of L moments to maximum river flows. The New Zealand
Statistician, 28, 2-10.

Pearson, C. P. (1995). Regional frequency analysis of low flows in New Zealand rivers. Journal
of Hydrology (NZ), 33, 94-122.

Pilon, P. J., and Adamowski, K. (1992). The value of regional information to flood frequency
analysis using the method of L-moments. Canadian Journal of Civil Engineering, 19, 137-147.

Potter, K. W., and Lettenmaier, D. P. (1990). A comparison of regional flood frequency
estimation methods using a resampling method. Water Resources Research, 26, 415-424.
Reed, D. W., and Stewart, E. J. (1994). Inter-site and inter-duration dependence of rainfall
extremes. In Statistics for the environment 2: water related issues, eds. V. Barnett and K. F.
Turkman, 125-143. New York: Wiley.

Rulli, M. C., and Rosso, R. (2002). An integrated simulation method for flash-flood risk
assessment: 1. Frequency predictions in the Bisagno River by combining stochastic and
deterministic methods. Hydrology and Earth System Sciences, 6, 267-283.

Schaefer, M. G. (1990). Regional analyses of precipitation annual maxima in Washington State.
Water Resources Research, 26, 119-131.

Smithers, J. C. (1996). Short-duration rainfall frequency model selection in Southern Africa.
Water S.A., 22, 211-217.

Smithers, J. C., and Schulze, R. E. (2001). A methodology for the estimation of short duration
design storms in South Africa using a regional approach based on L-moments. Journal of
Hydrology, 241, 42-52.

Vicente-Serrano, S. M., and Begueria-Portugues, S. (2003). Estimating extreme dry-spell risk in
the middle Ebro valley (northeastern Spain): a comparative analysis of partial duration series
with a general Pareto distribution and annual maxima series with a Gumbel distribution.
International Journal of Climatology, 23, 1103-1118.

Vogel, R. M., and Wilson, I. (1996). Probability distribution of annual maximum, mean, and
minimum streamflows in the United States. Journal of Hydraulic Engineering, 1, 69-76.

Wallis, J. R. (1993). Regional frequency studies using L-moments. In Concise encyclopedia of
environmental systems, ed. P. C. Young, 468-476. Oxford: Pergamon.

Waylen, P., and Laporte, M. S. (1999). Flooding and the El Nino-Southern Oscillation
phenomenon along the Pacific coast of Costa Rica. Hydrological Processes, 13, 2623-2638.

Waylen, P. R., and Zorn, M. R. (1998). Prediction of mean annual flows in north and central
Florida. Journal of the American Water Resources Association, 34, 149-157.

Zaidman, M. D., Keller, V., Young, A. R., and Cadman, D. (2003). Flow-duration-frequency
behaviour of British rivers based on annual minima data. Journal of Hydrology, 277, 195-213.

Zrinji, Z., and Burn, D. H. (1994). Flood frequency analysis for ungauged sites using a region of
influence approach. Journal of Hydrology, 153, 1-21.

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.

IBM software disclaimer
LMOMENTS: Fortran routines for use with the method of L-moments
Permission to use, copy, modify and distribute this software for any purpose and without fee is
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