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National Drought Atlas

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:

Equation

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

Equation

is a function of L-moments analogous to the coefficient of variation.


Standardized higher L-moments, defined by

Equation

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

Equation

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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