L-moment

probability weighted moments
In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution.wikipedia
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Kurtosis

excess kurtosisleptokurticplatykurtic
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).
Alternative measures of kurtosis are: the L-kurtosis, which is a scaled version of the fourth L-moment; measures based on four population or sample quantiles.

Mean absolute difference

MDMean differenceaverage absolute difference
The L-scale is equal to half the mean difference.
The mean absolute difference is twice the L-scale (the second L-moment), while the standard deviation is the square root of the variance about the mean (the second conventional central moment).

Moment (mathematics)

momentsmomentraw moment
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).
L-moment

Skewness

skewedskewskewed distribution
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).
Use of L-moments in place of moments provides a measure of skewness known as the L-skewness.

L-statistic

They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).
L-moment, L-statistic analogs of the conventional moments

Summary statistics

summary statisticSummarizationdata summarization
1) As summary statistics for data.
The Gini coefficient was originally developed to measure income inequality and is equivalent to one of the L-moments.

Higher-order statistics

high-orderhigh-order statisticshigher order statistics
This application shows the limited robustness of L-moments, i.e. L-statistics are not resistant statistics, as a single extreme value can throw them off, but because they are only linear (not higher-order statistics), they are less affected by extreme values than conventional moments.
An alternative to the use of HOS and higher moments is to instead use L-moments, which are linear statistics (linear combinations of order statistics), and thus more robust than HOS.

L-estimator

L-estimation
*L-estimator
Sample L-moments are L-estimators for the population L-moment, and have rather complex expressions.

Statistics

statisticalstatistical analysisstatistician
In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution.

Linear combination

linear combinationslinearly combined(finite) left ''R''-linear combinations
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).

Order statistic

order statisticsorderedth-smallest of items
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean). where X k:n denotes the k th order statistic (k th smallest value) in an independent sample of size n from the distribution of X and \mathrm{E} denotes expected value.

Standard deviation

standard deviationssample standard deviationsigma
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).

Mean

mean valuepopulation meanaverage
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).

Standardized moment

standardized central moments
analogous to standardized moments.

Independence (probability theory)

independentstatistically independentindependence
where X k:n denotes the k th order statistic (k th smallest value) in an independent sample of size n from the distribution of X and \mathrm{E} denotes expected value.

Sample (statistics)

samplesamplesstatistical sample
where X k:n denotes the k th order statistic (k th smallest value) in an independent sample of size n from the distribution of X and \mathrm{E} denotes expected value.

Expected value

expectationexpectedmean
where X k:n denotes the k th order statistic (k th smallest value) in an independent sample of size n from the distribution of X and \mathrm{E} denotes expected value.

Binomial transform

Euler transformEuler's transformEuler's transformation
Note that the coefficients of the k-th L-moment are the same as in the k-th term of the binomial transform, as used in the k-order finite difference (finite analog to the derivative).

Finite difference

difference operatorfinite differencesforward difference
Note that the coefficients of the k-th L-moment are the same as in the k-th term of the binomial transform, as used in the k-order finite difference (finite analog to the derivative).

Binomial coefficient

binomial coefficientschoosebinomials
The sample L-moments can be computed as the population L-moments of the sample, summing over r-element subsets of the sample hence averaging by dividing by the binomial coefficient:

Algorithm

algorithmscomputer algorithmalgorithm design
Sample L-moments can also be defined indirectly in terms of probability weighted moments, which leads to a more efficient algorithm for their computation.

Coefficient of variation

CVrelative standard deviationcoefficients of variation
A quantity analogous to the coefficient of variation, but based on L-moments, can also be defined:

Gini coefficient

Gini indexGiniequality
For a non-negative random variable, this lies in the interval (0,1) and is identical to the Gini coefficient.

Gumbel distribution

Gumbeldouble exponential distributionGumbel density
PWM are used to efficiently estimate the parameters of distributions expressable in inverse form such as the Gumbel, the Tukey, and the Wakeby distributions.

Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
In addition to doing these with standard moments, the latter (estimation) is more commonly done using maximum likelihood methods; however using L-moments provides a number of advantages.