Estimate L-Moments
Estimate the r'th L-moment from a random sample.
lMoment(x, r = 1, method = "unbiased",
plot.pos.cons = c(a = 0.35, b = 0), na.rm = FALSE)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
L-moment. 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: 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} ≤ 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 L-moment is given by:
λ_r = \frac{1}{r} ∑^{r-1}_{k=0} (-1)^k {r-1 \choose k} E[X_{r-k:r}]
for r = 1, 2, ….
Hosking (1990) shows that the above equation can be rewritten as:
λ_r = \int^1_0 x(u) P^*_{r-1}(u) du
where
P^*_r(u) = ∑^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:
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 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:
τ_r = \frac{λ_r}{λ_2}
for r = 2, 3, …. He shows that for a non-degenerate 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 L-moment analog of the coefficient of skewness, and the quantity
τ_4 = \frac{λ_4}{λ_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:
λ = \frac{λ_2}{λ_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:
λ_r = (-1)^{r-1} ∑^{r-1}_{k=0} p^*_{r-1,k} α_k = ∑^{r-1}_{j=0} p^*_{r-1,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
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} ∑^{r-1}_{k=0} p^*_{r-1,k} a_k = ∑^{r-1}_{j=0} p^*_{r-1,j} b_j
where
a_k = \frac{1}{n} ∑^{n-k}_{i=1} x_{i:n} \frac{{n-i \choose k}}{{n-1 \choose k}}
b_j = \frac{1}{n} ∑^{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{λ}_r = (-1)^{r-1} ∑^{r-1}_{k=0} p^*_{r-1,k} \tilde{α}_k = ∑^{r-1}_{j=0} p^*_{r-1,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 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{τ}_r = \frac{\tilde{λ}_r}{\tilde{λ}_2}
In particular, the L-moment skew is estimated by:
t_3 = \frac{l_3}{l_2}
or
\tilde{τ}_3 = \frac{\tilde{λ}_3}{\tilde{λ}_2}
and the L-moment kurtosis is estimated by:
t_4 = \frac{l_4}{l_2}
or
\tilde{τ}_4 = \frac{\tilde{λ}_4}{\tilde{λ}_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{λ} = \frac{\tilde{λ}_2}{\tilde{λ}_1}
A numeric scalar–the value of the r'th L-moment as defined by Hosking (1990).
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 on L-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.
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
# 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)
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