L-moment

In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution.^[1][2][3][4] 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). Standardised L-moments are called L-moment ratios and are analogous to standardized moments. Just as for conventional moments, a theoretical distribution has a set of population L-moments. Sample L-moments can be defined for a sample from the population, and can be used as estimators of the population L-moments.

Population L-moments

For a random variable X, the rth population L-moment is^[1]

${\displaystyle \lambda _{r}={\frac {\ 1\ }{r}}\sum _{k=0}^{r-1}(-1)^{k}{\binom {r-1}{k}}\operatorname {\mathbb {E} } \{\ X_{r-k:r}\ \}\ ,}$

where X_k:n denotes the rth

of size n from the distribution of X and ${\displaystyle \ \mathbb {E} \ }$ denotes expected value operator. In particular, the first four population L-moments are

${\displaystyle \lambda _{1}=\operatorname {\mathbb {E} } \,\!\{\ X\ \}}$

${\displaystyle \lambda _{2}={\frac {\ 1\ }{2}}{\Bigl (}\ \operatorname {\mathbb {E} } \,\!\{\ X_{2:2}\ \}-\operatorname {\mathbb {E} } \,\!\{\ X_{1:2}\ \}\ {\Bigr )}}$

${\displaystyle \lambda _{3}={\frac {\ 1\ }{3}}{\Bigl (}\ \operatorname {\mathbb {E} } \,\!\{\ X_{3:3}\ \}-2\operatorname {\mathbb {E} } \,\!\{\ X_{2:3}\ \}+\operatorname {\mathbb {E} } \,\!\{\ X_{1:3}\ \}\ {\Bigr )}}$

${\displaystyle \lambda _{4}={\frac {\ 1\ }{4}}{\Bigl (}\ \operatorname {\mathbb {E} } \,\!\{\ X_{4:4}\ \}-3\operatorname {\mathbb {E} } \,\!\{\ X_{3:4}\ \}+3\operatorname {\mathbb {E} } \,\!\{\ X_{2:4}\ \}-\operatorname {\mathbb {E} } \,\!\{\ X_{1:4}\ \}\ {\Bigr )}~.}$

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

The first two of these L-moments have conventional names:

${\displaystyle \ \lambda _{1}\ }$ is the "mean", "L-mean", or "L-location",

${\displaystyle \ \lambda _{2}\ }$ is the "L-scale".

The L-scale is equal to half the Mean absolute difference.^[5]

Sample L-moments

The sample L-moments can be computed as the population L-moments of the sample, summing over r-element subsets of the sample ${\displaystyle \left\{x_{1}<\cdots <x_{j}<\cdots <x_{r}\right\},}$ hence averaging by dividing by the binomial coefficient:

${\displaystyle \lambda _{r}={\frac {1}{\ r\cdot {\tbinom {n}{r}}\ }}\ \sum _{x_{1}<\cdots <x_{j}<\cdots <x_{r}}\ (-1)^{r-j}{\binom {r-1}{j}}\ x_{j}~.}$

Grouping these by order statistic counts the number of ways an element of an n element sample can be the jth element of an r element subset, and yields formulas of the form below. Direct estimators for the first four L-moments in a finite sample of n observations are:^[6]

${\displaystyle \ell _{1}={\frac {1}{\ {\tbinom {n}{1}}\ }}\sum _{i=1}^{n}\ x_{(i)}\ }$

${\displaystyle \ell _{2}={\frac {1}{\ 2\cdot {\tbinom {n}{2}}\ }}\sum _{i=1}^{n}\ {\Bigl [}\ {\tbinom {i-1}{1}}-{\tbinom {n-i}{1}}\ {\Bigr ]}\ x_{(i)}\ }$

${\displaystyle \ell _{3}={\frac {1}{\ 3\cdot {\tbinom {n}{3}}\ }}\sum _{i=1}^{n}\ {\Bigl [}\ {\tbinom {i-1}{2}}-2{\tbinom {i-1}{1}}{\tbinom {n-i}{1}}+{\tbinom {n-i}{2}}\ {\Bigr ]}\ x_{(i)}\ }$

${\displaystyle \ell _{4}={\frac {1}{\ 4\cdot {\tbinom {n}{4}}\ }}\sum _{i=1}^{n}\ {\Bigl [}\ {\tbinom {i-1}{3}}-3{\tbinom {i-1}{2}}{\tbinom {n-i}{1}}+3{\tbinom {i-1}{1}}{\tbinom {n-i}{2}}-{\tbinom {n-i}{3}}\ {\Bigr ]}\ x_{(i)}\ }$

where x_(i) is the ith order statistic and ${\displaystyle \ {\tbinom {\boldsymbol {\cdot }}{\boldsymbol {\cdot }}}\ }$ is a binomial coefficient. Sample L-moments can also be defined indirectly in terms of probability weighted moments,^[1][7][8] which leads to a more efficient algorithm for their computation.^[6][9]

L-moment ratios

A set of L-moment ratios, or scaled L-moments, is defined by

${\displaystyle \tau _{r}=\lambda _{r}/\lambda _{2},\qquad r=3,4,\dots ~.}$

The most useful of these are ${\displaystyle \ \tau _{3}\ ,}$ called the L-skewness, and ${\displaystyle \ \tau _{4}\ ,}$ the L-kurtosis.

L-moment ratios lie within the interval  ( −1, 1 ) . Tighter bounds can be found for some specific L-moment ratios; in particular, the L-kurtosis ${\displaystyle \ \tau _{4}\ }$ lies in  [ −+ 1 /4, 1 ) , and

${\displaystyle \ {\tfrac {\ 1\ }{4}}\left(\ 5\ \tau _{3}^{2}-1\ \right)\leq \tau _{4}<1~.}$^[1]

A quantity analogous to the coefficient of variation, but based on L-moments, can also be defined: ${\displaystyle \ \tau =\lambda _{2}/\lambda _{1}\ ,}$ which is called the "coefficient of L-variation", or "L-CV". For a non-negative random variable, this lies in the interval   ( 0, 1 )  ^[1] and is identical to the Gini coefficient.^[10]

Related quantities

L-moments are statistical quantities that are derived from probability weighted moments^[11] (PWM) which were defined earlier (1979).^[7] PWM are used to efficiently estimate the parameters of distributions expressable in inverse form such as the Gumbel,^[8] the Tukey lambda, and the Wakeby distributions.

Usage

There are two common ways that L-moments are used, in both cases analogously to the conventional moments:

1. As summary statistics for data.
2. To derive estimators for the parameters of
   probability distributions, applying the method of moments
   to the L-moments rather than conventional moments.

In addition to doing these with standard moments, the latter (estimation) is more commonly done using

than conventional moments, and existence of higher L-moments only requires that the random variable have finite mean. One disadvantage of L-moment ratios for estimation is their typically smaller sensitivity. For instance, the Laplace distribution has a kurtosis of 6 and weak exponential tails, but a larger 4th L-moment ratio than e.g. the student-t distribution with d.f.=3, which has an infinite kurtosis and much heavier tails.

As an example consider a dataset with a few data points and one outlying data value. If the ordinary

), they are less affected by extreme values than conventional moments.

Another advantage L-moments have over conventional moments is that their existence only requires the random variable to have finite mean, so the L-moments exist even if the higher conventional moments do not exist (for example, for

Some appearances of L-moments in the statistical literature include the book by David & Nagaraja (2003, Section 9.9)[12] and a number of papers.^{[10][13][14][15][16][17]} A number of favourable comparisons of L-moments with ordinary moments have been reported.^[18][19]

Values for some common distributions

The table below gives expressions for the first two L moments and numerical values of the first two L-moment ratios of some common

More complex expressions have been derived for some further distributions for which the L-moment ratios vary with one or more of the distributional parameters, including the log-normal, Gamma, generalized Pareto, generalized extreme value, and generalized logistic distributions.^[1]

Distribution	Parameters	mean, λ1	L-scale, λ2	L-skewness, τ3	L-kurtosis, τ4
Uniform	a, b	1 /2(a + b)	1 /6(b – a)	0	0
Logistic	μ, s	μ	s	0	1 /6 = 0.1667
Normal	μ, σ2	μ	σ/ √π	0	30 θm/π  - 9 = 0.1226
Laplace	μ, b	μ	3 / 4 b	0	1/ 3 √2  = 0.2357
Student's t, 2 d.f.	ν = 2	0	π/ 2 √2  = 1.111	0	3 / 8 = 0.375
Student's t, 4 d.f.	ν = 4	0	15 /64 π = 0.7363	0	111 /512 = 0.2168
Exponential	λ	1/ λ	1/ 2 λ	1 /3 = 0.3333	1 /6 = 0.1667
Gumbel	μ, β	μ + γe β	β log2(3)	2 log2(3) - 3 = 0.1699	16 - 10 log2(3) = 0.1504

The notation for the parameters of each distribution is the same as that used in the linked article. In the expression for the mean of the

0.5772 1566 4901 ... .

Extensions

Trimmed L-moments are generalizations of L-moments that give zero weight to extreme observations. They are therefore more robust to the presence of outliers, and unlike L-moments they may be well-defined for distributions for which the mean does not exist, such as the Cauchy distribution.^[20]

See also

• L-estimator

References

External links

• The L-moments page Jonathan R.M. Hosking, IBM Research
• L Moments. Dataplot reference manual, vol. 1, auxiliary chapter. National Institute of Standards and Technology, 2006. Accessed 2010-05-25.
• Lmo lightweight Python includes functions for fast calculation of L-moments, trimmed L-moments, and multivariate L-comoments.