# 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

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**

### 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**

### Mean

**mean valuepopulation meanaverage**

### 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.