| lMoment {EnvStats} | R Documentation |
Estimate L-Moments
Description
Estimate the r'th L-moment from a random sample.
Usage
lMoment(x, r = 1, method = "unbiased",
plot.pos.cons = c(a = 0.35, b = 0), na.rm = FALSE)
Arguments
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
|
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 |
Details
Definitions: L-Moments and L-Moment Ratios
The definition of an L-moment 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} \le x_{2:n} \le \ldots \le x_{n:n}
denote the order statistics of a random sample of size n drawn from the
distribution of X. Then the r'th L-moment is given by:
\lambda_r = \frac{1}{r} \sum^{r-1}_{k=0} (-1)^k {r-1 \choose k} E[X_{r-k:r}]
for r = 1, 2, \ldots.
Hosking (1990) shows that the above equation can be rewritten as:
\lambda_r = \int^1_0 x(u) P^*_{r-1}(u) du
where
P^*_r(u) = \sum^r_{k=0} p^*_{r,k} u^k
p^*_{r,k} = (-1)^{r-k} {r \choose k} {r+k \choose k} = \frac{(-1)^{r-k} (r+k)!}{(k!)^2 (r-k)!}
The first four L-moments are given by:
\lambda_1 = E[X]
\lambda_2 = \frac{1}{2} E[X_{2:2} - X_{1:2}]
\lambda_3 = \frac{1}{3} E[X_{3:3} - 2X_{2:3} + X_{1:3}]
\lambda_4 = \frac{1}{4} E[X_{4:4} - 3X_{3:4} + 3X_{2:4} - X_{1:4}]
Thus, the first L-moment is a measure of location, and the second
L-moment is a measure of scale.
Hosking (1990) defines the L-moment ratios of X to be:
\tau_r = \frac{\lambda_r}{\lambda_2}
for r = 2, 3, \ldots. He shows that for a non-degenerate random variable
with a finite mean, these quantities lie in the interval (-1, 1).
The quantity
\tau_3 = \frac{\lambda_3}{\lambda_2}
is the L-moment analog of the coefficient of skewness, and the quantity
\tau_4 = \frac{\lambda_4}{\lambda_2}
is the L-moment analog of the coefficient of kurtosis. Hosking (1990) also
defines an L-moment analog of the coefficient of variation (denoted the
L-CV) as:
\lambda = \frac{\lambda_2}{\lambda_1}
He shows that for a positive-valued random variable, the L-CV lies
in the interval (0, 1).
Relationship Between L-Moments and Probability-Weighted Moments
Hosking (1990) and Hosking and Wallis (1995) show that L-moments can be
written as linear combinations of probability-weighted moments:
\lambda_r = (-1)^{r-1} \sum^{r-1}_{k=0} p^*_{r-1,k} \alpha_k = \sum^{r-1}_{j=0} p^*_{r-1,j} \beta_j
where
\alpha_k = M(1, 0, k) = \frac{1}{k+1} E[X_{1:k+1}]
\beta_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
probability-weighted moments.
Estimating L-Moments
The two commonly used methods for estimating L-moments are the
“unbiased” method based on U-statistics (Hoeffding, 1948;
Lehmann, 1975, pp. 362-371), and the “plotting-position” method.
Hosking and Wallis (1995) recommend using the unbiased method for almost all
applications.
Unbiased Estimators (method="unbiased")
Using the relationship between L-moments and probability-weighted moments
explained above, the unbiased estimator of the r'th L-moment is based on
unbiased estimators of probability-weighted moments and is given by:
l_r = (-1)^{r-1} \sum^{r-1}_{k=0} p^*_{r-1,k} a_k = \sum^{r-1}_{j=0} p^*_{r-1,j} b_j
where
a_k = \frac{1}{n} \sum^{n-k}_{i=1} x_{i:n} \frac{{n-i \choose k}}{{n-1 \choose k}}
b_j = \frac{1}{n} \sum^{n}_{i=j+1} x_{i:n} \frac{{i-1 \choose j}}{{n-1 \choose j}}
Plotting-Position Estimators (method="plotting.position")
Using the relationship between L-moments and probability-weighted moments
explained above, the plotting-position estimator of the r'th L-moment
is based on the plotting-position estimators of probability-weighted moments and
is given by:
\tilde{\lambda}_r = (-1)^{r-1} \sum^{r-1}_{k=0} p^*_{r-1,k} \tilde{\alpha}_k = \sum^{r-1}_{j=0} p^*_{r-1,j} \tilde{\beta}_j
where
\tilde{\alpha}_k = \frac{1}{n} \sum^n_{i=1} (1 - p_{i:n})^k x_{i:n}
\tilde{\beta}_j = \frac{1}{n} \sum^{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 distribution-free estimate of the cdf of
X evaluated at the i'th order statistic. Typically, plotting
positions have the form:
p_{i:n} = \frac{i-a}{n+b}
where b > -a > -1. For this form of plotting position, the
plotting-position estimators are asymptotically equivalent to their
unbiased estimator counterparts.
Estimating L-Moment Ratios
L-moment ratios are estimated by simply replacing the population
L-moments with the estimated L-moments. 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 plotting-position estimators are given by:
\tilde{\tau}_r = \frac{\tilde{\lambda}_r}{\tilde{\lambda}_2}
In particular, the L-moment skew is estimated by:
t_3 = \frac{l_3}{l_2}
or
\tilde{\tau}_3 = \frac{\tilde{\lambda}_3}{\tilde{\lambda}_2}
and the L-moment kurtosis is estimated by:
t_4 = \frac{l_4}{l_2}
or
\tilde{\tau}_4 = \frac{\tilde{\lambda}_4}{\tilde{\lambda}_2}
Similarly, the L-moment coefficient of variation can be estimated using
the unbiased L-moment estimators:
l = \frac{l_2}{l_1}
or using the plotting-position L-moment estimators:
\tilde{\lambda} = \frac{\tilde{\lambda}_2}{\tilde{\lambda}_1}
Value
A numeric scalar–the value of the r'th L-moment as defined by Hosking (1990).
Note
Hosking (1990) introduced the idea of L-moments, 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 L-moments parallels the
theory of conventional moments. L-moments have several advantages over
conventional moments, including:
-
L-moments can characterize a wider range of distributions because they always exist as long as the distribution has a finite mean. -
L-moments are estimated by linear combinations of order statistics, so estimators based onL-moments are more robust to the presence of outliers than estimators based on conventional moments. Based on the author's and others' experience,
L-moment estimators are less biased and approximate their asymptotic distribution more closely in finite samples than estimators based on conventional moments.-
L-moment estimators are sometimes more efficient (smaller RMSE) than even maximum likelihood estimators for small samples.
Hosking (1990) presents a table with formulas for the L-moments of common
probability distributions. Articles that illustrate the use of L-moments
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
probabiity-weighted moments and L-moments.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Fill, H.D., and J.R. Stedinger. (1995). L Moment and Probability Plot
Correlation Coefficient Goodness-of-Fit Tests for the Gumbel Distribution and
Impact of Autocorrelation. Water Resources Research 31(1), 225–229.
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(1), 105–124.
Hosking, J.R.M., and J.R. Wallis (1995). A Comparison of Unbiased and
Plotting-Position 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.
See Also
cv, skewness, kurtosis,
pwMoment.
Examples
# Generate 20 observations from a generalized extreme value distribution
# with parameters location=10, scale=2, and shape=.25, then compute the
# first four L-moments.
# (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 L-moments based on the plotting-position 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)