Methodology
L-moments
The Atlas used a then relatively new method called "L-moments" to
estimate population distributions from sample data sets. This was one of
the first uses of L-moments on large data sets, but the method is now
being used widely by researchers, including National Oceanic and
Atmospheric Administration meteorologists. The "L" stands for a
linear combination of order statistics. This method has been shown to
provide more reliable population estimates from small sample sizes because
it reduces the influence that one outlier has on the selection of the
population type and parameters.
The order statistics of a random sample of size n are the sample
values arranged in ascending order: X1:n X2:n Xn:n. L-moments (Hosking, 1990)
are certain linear combinations of the order statistics from small samples
that can be used to summarize the sample and the distribution from which
the sample was drawn.
The first four L-moments are the following expected values of linear
combinations:
The first L-moment is the mean of the distribution. The second L-moment
is a measure of dispersion, analogous to, but not equal to, the standard
deviation. The L-CV, defined by
is a function of L-moments analogous to the coefficient of variation.
Standardized higher L-moments, defined by
include the L-skewness, τ3, and the L-kurtosis, τ4.
As their names imply, these are measures of the skewness and kurtosis of
the distribution.
When estimated from a sample of size n, L-moments are most
conveniently calculated by first calculating the quantities
The sample L-moments are then calculated by
l1 = b0 ,
l2 = 2b1 - b0 ,
l3 = 6b2 - 6b1 + b0 ,
l4 = 20b3 -
30b2 + 12b1 - b0;
lr is the sample estimate of λr .
Similarly, the ratios τ, τ3,
and τ4 are estimated
respectively by
t =l 2 /l1, t3 =l3 /l2, t4 =l4 /l2.
Ordinarily, four L-moments are calculated, giving measures of location,
dispersion, skewness and kurtosis. Because L-moments involve only linear
combinations of the data, and do not require raising the data values to
higher powers, they are less sensitive than the conventional moments to
the numerical values of the most extreme observations. This and other
advantages of L-moments have been demonstrated by several authors (Hosking,
1990, 1992; Royston, 1992; Vogel and Fennessy, 1993). Guttman (1994) has
discussed the sensitivity of sample L-moments to the size of the sample.
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