Description Usage Arguments Details Value Note Author(s) References See Also Examples
Estimate the r'th Lmoment from a random sample.
1 2 
x 
numeric vector of observations. 
r 
positive integer specifying the order of the moment. 
method 
character string specifying what method to use to compute the
Lmoment. The possible values are 
plot.pos.cons 
numeric vector of length 2 specifying the constants used in the formula for the
plotting positions when 
na.rm 
logical scalar indicating whether to remove missing values from 
Definitions: LMoments and LMoment Ratios
The definition of an Lmoment given by Hosking (1990) is as follows.
Let X denote a random variable with cdf F, and let x(p)
denote the p'th quantile of the distribution. Furthermore, let
x_{1:n} ≤ x_{2:n} ≤ … ≤ x_{n:n}
denote the order statistics of a random sample of size n drawn from the distribution of X. Then the r'th Lmoment is given by:
λ_r = \frac{1}{r} ∑^{r1}_{k=0} (1)^k {r1 \choose k} E[X_{rk:r}]
for r = 1, 2, ….
Hosking (1990) shows that the above equation can be rewritten as:
λ_r = \int^1_0 x(u) P^*_{r1}(u) du
where
P^*_r(u) = ∑^r_{k=0} p^*_{r,k} u^k
p^*_{r,k} = (1)^{rk} {r \choose k} {r+k \choose k} = \frac{(1)^{rk} (r+k)!}{(k!)^2 (rk)!}
The first four Lmoments are given by:
λ_1 = E[X]
λ_2 = \frac{1}{2} E[X_{2:2}  X_{1:2}]
λ_3 = \frac{1}{3} E[X_{3:3}  2X_{2:3} + X_{1:3}]
λ_4 = \frac{1}{4} E[X_{4:4}  3X_{3:4} + 3X_{2:4}  X_{1:4}]
Thus, the first Lmoment is a measure of location, and the second Lmoment is a measure of scale.
Hosking (1990) defines the Lmoment ratios of X to be:
τ_r = \frac{λ_r}{λ_2}
for r = 2, 3, …. He shows that for a nondegenerate random variable with a finite mean, these quantities lie in the interval (1, 1). The quantity
τ_3 = \frac{λ_3}{λ_2}
is the Lmoment analog of the coefficient of skewness, and the quantity
τ_4 = \frac{λ_4}{λ_2}
is the Lmoment analog of the coefficient of kurtosis. Hosking (1990) also defines an Lmoment analog of the coefficient of variation (denoted the LCV) as:
λ = \frac{λ_2}{λ_1}
He shows that for a positivevalued random variable, the LCV lies in the interval (0, 1).
Relationship Between LMoments and ProbabilityWeighted Moments
Hosking (1990) and Hosking and Wallis (1995) show that Lmoments can be
written as linear combinations of probabilityweighted moments:
λ_r = (1)^{r1} ∑^{r1}_{k=0} p^*_{r1,k} α_k = ∑^{r1}_{j=0} p^*_{r1,j} β_j
where
α_k = M(1, 0, k) = \frac{1}{k+1} E[X_{1:k+1}]
β_j = M(1, j, 0) = \frac{1}{j+1} E[X_{j+1:j+1}]
See the help file for pwMoment
for more information on
probabilityweighted moments.
Estimating LMoments
The two commonly used methods for estimating Lmoments are the
“unbiased” method based on Ustatistics (Hoeffding, 1948;
Lehmann, 1975, pp. 362371), and the “plottingposition” method.
Hosking and Wallis (1995) recommend using the unbiased method for almost all
applications.
Unbiased Estimators (method="unbiased"
)
Using the relationship between Lmoments and probabilityweighted moments
explained above, the unbiased estimator of the r'th Lmoment is based on
unbiased estimators of probabilityweighted moments and is given by:
l_r = (1)^{r1} ∑^{r1}_{k=0} p^*_{r1,k} a_k = ∑^{r1}_{j=0} p^*_{r1,j} b_j
where
a_k = \frac{1}{n} ∑^{nk}_{i=1} x_{i:n} \frac{{ni \choose k}}{{n1 \choose k}}
b_j = \frac{1}{n} ∑^{n}_{i=j+1} x_{i:n} \frac{{i1 \choose j}}{{n1 \choose j}}
PlottingPosition Estimators (method="plotting.position"
)
Using the relationship between Lmoments and probabilityweighted moments
explained above, the plottingposition estimator of the r'th Lmoment
is based on the plottingposition estimators of probabilityweighted moments and
is given by:
\tilde{λ}_r = (1)^{r1} ∑^{r1}_{k=0} p^*_{r1,k} \tilde{α}_k = ∑^{r1}_{j=0} p^*_{r1,j} \tilde{β}_j
where
\tilde{α}_k = \frac{1}{n} ∑^n_{i=1} (1  p_{i:n})^k x_{i:n}
\tilde{β}_j = \frac{1}{n} ∑^{n}_{i=1} p^j_{i:n} x_{i:n}
and
p_{i:n} = \hat{F}(x_{i:n})
denotes the plotting position of the i'th order statistic in the random sample of size n, that is, a distributionfree estimate of the cdf of X evaluated at the i'th order statistic. Typically, plotting positions have the form:
p_{i:n} = \frac{ia}{n+b}
where b > a > 1. For this form of plotting position, the plottingposition estimators are asymptotically equivalent to their unbiased estimator counterparts.
Estimating LMoment Ratios
Lmoment ratios are estimated by simply replacing the population
Lmoments with the estimated Lmoments. The estimated ratios
based on the unbiased estimators are given by:
t_r = \frac{l_r}{l_2}
and the estimated ratios based on the plottingposition estimators are given by:
\tilde{τ}_r = \frac{\tilde{λ}_r}{\tilde{λ}_2}
In particular, the Lmoment skew is estimated by:
t_3 = \frac{l_3}{l_2}
or
\tilde{τ}_3 = \frac{\tilde{λ}_3}{\tilde{λ}_2}
and the Lmoment kurtosis is estimated by:
t_4 = \frac{l_4}{l_2}
or
\tilde{τ}_4 = \frac{\tilde{λ}_4}{\tilde{λ}_2}
Similarly, the Lmoment coefficient of variation can be estimated using the unbiased Lmoment estimators:
l = \frac{l_2}{l_1}
or using the plottingposition Lmoment estimators:
\tilde{λ} = \frac{\tilde{λ}_2}{\tilde{λ}_1}
A numeric scalar–the value of the r'th Lmoment as defined by Hosking (1990).
Hosking (1990) introduced the idea of Lmoments, which are expectations of certain linear combinations of order statistics, as the basis of a general theory of describing theoretical probability distributions, computing summary statistics from observed data, estimating distribution parameters and quantiles, and performing hypothesis tests. The theory of Lmoments parallels the theory of conventional moments. Lmoments have several advantages over conventional moments, including:
Lmoments can characterize a wider range of distributions because they always exist as long as the distribution has a finite mean.
Lmoments are estimated by linear combinations of order statistics, so estimators based on Lmoments are more robust to the presence of outliers than estimators based on conventional moments.
Based on the author's and others' experience, Lmoment estimators are less biased and approximate their asymptotic distribution more closely in finite samples than estimators based on conventional moments.
Lmoment estimators are sometimes more efficient (smaller RMSE) than even maximum likelihood estimators for small samples.
Hosking (1990) presents a table with formulas for the Lmoments of common probability distributions. Articles that illustrate the use of Lmoments include Fill and Stedinger (1995), Hosking and Wallis (1995), and Vogel and Fennessey (1993).
Hosking (1990) and Hosking and Wallis (1995) show the relationship between probabiityweighted moments and Lmoments.
Steven P. Millard ([email protected])
Fill, H.D., and J.R. Stedinger. (1995). L Moment and Probability Plot Correlation Coefficient GoodnessofFit Tests for the Gumbel Distribution and Impact of Autocorrelation. Water Resources Research 31(1), 225–229.
Hosking, J.R.M. (1990). LMoments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society, Series B 52(1), 105–124.
Hosking, J.R.M., and J.R. Wallis (1995). A Comparison of Unbiased and PlottingPosition Estimators of L Moments. Water Resources Research 31(8), 2019–2025.
Vogel, R.M., and N.M. Fennessey. (1993). L Moment Diagrams Should Replace Product Moment Diagrams. Water Resources Research 29(6), 1745–1752.
cv
, skewness
, kurtosis
,
pwMoment
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  # Generate 20 observations from a generalized extreme value distribution
# with parameters location=10, scale=2, and shape=.25, then compute the
# first four Lmoments.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat < rgevd(20, location = 10, scale = 2, shape = 0.25)
lMoment(dat)
#[1] 10.59556
lMoment(dat, 2)
#[1] 1.0014
lMoment(dat, 3)
#[1] 0.1681165
lMoment(dat, 4)
#[1] 0.08732692
#
# Now compute some Lmoments based on the plottingposition estimators:
lMoment(dat, method = "plotting.position")
#[1] 10.59556
lMoment(dat, 2, method = "plotting.position")
#[1] 1.110264
lMoment(dat, 3, method="plotting.position", plot.pos.cons = c(.325,1))
#[1] 0.4430792
#
# Clean up
#
rm(dat)

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