# 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.