lMoment {EnvStats} | R Documentation |
Estimate
-Moments
Description
Estimate the 'th
-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: -Moments and
-Moment Ratios
The definition of an -moment given by Hosking (1990) is as follows.
Let
denote a random variable with cdf
, and let
denote the
'th quantile of the distribution. Furthermore, let
denote the order statistics of a random sample of size drawn from the
distribution of
. Then the
'th
-moment is given by:
for .
Hosking (1990) shows that the above equation can be rewritten as:
where
The first four -moments are given by:
Thus, the first -moment is a measure of location, and the second
-moment is a measure of scale.
Hosking (1990) defines the -moment ratios of
to be:
for . He shows that for a non-degenerate random variable
with a finite mean, these quantities lie in the interval
.
The quantity
is the -moment analog of the coefficient of skewness, and the quantity
is the -moment analog of the coefficient of kurtosis. Hosking (1990) also
defines an
-moment analog of the coefficient of variation (denoted the
-CV) as:
He shows that for a positive-valued random variable, the -CV lies
in the interval
.
Relationship Between -Moments and Probability-Weighted Moments
Hosking (1990) and Hosking and Wallis (1995) show that -moments can be
written as linear combinations of probability-weighted moments:
where
See the help file for pwMoment
for more information on
probability-weighted moments.
Estimating L-Moments
The two commonly used methods for estimating -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 -moments and probability-weighted moments
explained above, the unbiased estimator of the
'th
-moment is based on
unbiased estimators of probability-weighted moments and is given by:
where
Plotting-Position Estimators (method="plotting.position"
)
Using the relationship between -moments and probability-weighted moments
explained above, the plotting-position estimator of the
'th
-moment
is based on the plotting-position estimators of probability-weighted moments and
is given by:
where
and
denotes the plotting position of the 'th order statistic in the random
sample of size
, that is, a distribution-free estimate of the cdf of
evaluated at the
'th order statistic. Typically, plotting
positions have the form:
where . For this form of plotting position, the
plotting-position estimators are asymptotically equivalent to their
unbiased estimator counterparts.
Estimating -Moment Ratios
-moment ratios are estimated by simply replacing the population
-moments with the estimated
-moments. The estimated ratios
based on the unbiased estimators are given by:
and the estimated ratios based on the plotting-position estimators are given by:
In particular, the -moment skew is estimated by:
or
and the -moment kurtosis is estimated by:
or
Similarly, the -moment coefficient of variation can be estimated using
the unbiased
-moment estimators:
or using the plotting-position L-moment estimators:
Value
A numeric scalar–the value of the 'th
-moment as defined by Hosking (1990).
Note
Hosking (1990) introduced the idea of -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
-moments parallels the
theory of conventional moments.
-moments have several advantages over
conventional moments, including:
-
-moments can characterize a wider range of distributions because they always exist as long as the distribution has a finite mean.
-
-moments are estimated by linear combinations of order statistics, so estimators based on
-moments are more robust to the presence of outliers than estimators based on conventional moments.
Based on the author's and others' experience,
-moment estimators are less biased and approximate their asymptotic distribution more closely in finite samples than estimators based on conventional moments.
-
-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 -moments of common
probability distributions. Articles that illustrate the use of
-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 -moments.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Fill, H.D., and J.R. Stedinger. (1995). 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 Moments. Water Resources Research
31(8), 2019–2025.
Vogel, R.M., and N.M. Fennessey. (1993). 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)