# L-moment

Short description: Statistical sequence characterizing probability distributions

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 Xk:n denotes the rth order statistic (rth smallest value) in an independent sample 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 $\displaystyle{ \left\{ x_1 \lt \cdots \lt x_j \lt \cdots \lt x_r \right\}, }$ hence averaging by dividing by the binomial coefficient:

$\displaystyle{ \lambda_r = \frac{1}{\ r\cdot\tbinom{n}{r}\ }\ \sum_{x_1 \lt \cdots \lt x_j \lt \cdots \lt x_r}\ (-1)^{r-j} \binom{r-1}{j}\ x_j ~. }$

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

$\displaystyle{ \ \tfrac{\ 1\ }{4} \left(\ 5\ \tau_3^2 - 1\ \right) \leq \tau_4 \lt 1 ~. }$[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 maximum likelihood methods; however using L-moments provides a number of advantages. Specifically, L-moments are more robust 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 standard deviation of this data set is taken it will be highly influenced by this one point: however, if the L-scale is taken it will be far less sensitive to this data value. Consequently, L-moments are far more meaningful when dealing with outliers in data than conventional moments. However, there are also other better suited methods to achieve an even higher robustness than just replacing moments by L-moments. One example of this is using L-moments as summary statistics in extreme value theory (EVT). 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.

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 Student's t distribution with low degrees of freedom). A finite variance is required in addition in order for the standard errors of estimates of the L-moments to be finite.[1]

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 continuous probability distributions with constant L-moment ratios.[1][5] 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(ba) 0 0
Logistic μ, s μ s 0 0.1667  1 /6 = 0.1667

Normal μ, σ2 μ σ/π 0 0.1226 30 θm/π  - 9 = 0.1226

Laplace μ, b μ  3 / 4 b 0 0.2357 1/ 3 2 = 0.2357

Student's t, 2 d.f. ν = 2 0 π/ 2 2 = 1.111 0 0.375  3 / 8 = 0.375

Student's t, 4 d.f. ν = 4 0  15 /64 π = 0.7363 0 0.2168  111 /512 = 0.2168

Exponential λ 1/λ 1/ 2 λ 0.3333  1 /3 = 0.3333

0.1667  1 /6 = 0.1667

Gumbel μ, β μ + γe β β log2(3) 0.1699 2 log2(3) - 3 = 0.1699

0.1504 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 Gumbel distribution, γe is the Euler–Mascheroni constant 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]

## References

1. Hosking, J.R.M. (1990). "L-moments: analysis and estimation of distributions using linear combinations of order statistics". Journal of the Royal Statistical Society, Series B 52 (1): 105–124.
2. Hosking, J.R.M. (1992). "Moments or L moments? An example comparing two measures of distributional shape". The American Statistician 46 (3): 186–189. doi:10.2307/2685210.
3. Hosking, J.R.M. (2006). "On the characterization of distributions by their L-moments". Journal of Statistical Planning and Inference 136: 193–198. doi:10.1016/j.jspi.2004.06.004.
4. Asquith, W.H. (2011) Distributional analysis with L-moment statistics using the R environment for statistical computing, Create Space Independent Publishing Platform, [print-on-demand], ISBN:1-463-50841-7
5. Jones, M.C. (2002). "Student's simplest distribution". Journal of the Royal Statistical Society, Series D 51 (1): 41–49. doi:10.1111/1467-9884.00297.
6. Wang, Q.J. (1996). "Direct sample estimators of L-moments". Water Resources Research 32 (12): 3617–3619. doi:10.1029/96WR02675.
7. Greenwood, J.A.; Landwehr, J.M.; Matalas, N.C.; Wallis, J.R. (1979). "Probability weighted moments: Definition and relation to parameters of several distributions expressed in inverse form". Water Resources Research 15 (5): 1049–1054. doi:10.1029/WR015i005p01049.
8. Landwehr, J.M.; Matalas, N.C.; Wallis, J.R. (1979). "Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles". Water Resources Research 15 (5): 1055–1064. doi:10.1029/WR015i005p01055.
9. Valbuena, R.; Maltamo, M.; Mehtätalo, L.; Packalen, P. (2017). "Key structural features of Boreal forests may be detected directly using L-moments from airborne lidar data". Remote Sensing of Environment 194: 437–446. doi:10.1016/j.rse.2016.10.024.
10. Hosking, JRM; Wallis, JR (2005). Regional Frequency Analysis: An Approach Based on L-moments. Cambridge University Press. pp. 3. ISBN 978-0521019408. Retrieved 22 January 2013.
11. David, H. A.; Nagaraja, H. N. (2003). Order Statistics (3rd ed.). Wiley. ISBN 978-0-471-38926-2.
12. Serfling, R.; Xiao, P. (2007). "A contribution to multivariate L-moments: L-comoment matrices". Journal of Multivariate Analysis 98 (9): 1765–1781. doi:10.1016/j.jmva.2007.01.008.
13. Delicado, P.; Goria, M. N. (2008). "A small sample comparison of maximum likelihood, moments and L-moments methods for the asymmetric exponential power distribution". Computational Statistics & Data Analysis 52 (3): 1661–1673. doi:10.1016/j.csda.2007.05.021.
14. Alkasasbeh, M. R.; Raqab, M. Z. (2009). "Estimation of the generalized logistic distribution parameters: comparative study". Statistical Methodology 6 (3): 262–279. doi:10.1016/j.stamet.2008.10.001.
15. Jones, M. C. (2004). "On some expressions for variance, covariance, skewness and L-moments". Journal of Statistical Planning and Inference 126 (1): 97–106. doi:10.1016/j.jspi.2003.09.001.
16. Jones, M. C. (2009). "Kumaraswamy's distribution: A beta-type distribution with some tractability advantages". Statistical Methodology 6 (1): 70–81. doi:10.1016/j.stamet.2008.04.001.
17. Royston, P. (1992). "Which measures of skewness and kurtosis are best?". Statistics in Medicine 11 (3): 333–343. doi:10.1002/sim.4780110306. PMID 1609174.
18. Ulrych, T. J.; Velis, D. R.; Woodbury, A. D.; Sacchi, M. D. (2000). "L-moments and C-moments". Stochastic Environmental Research and Risk Assessment 14 (1): 50–68. doi:10.1007/s004770050004.
19. Elamir, Elsayed A. H.; Seheult, Allan H. (2003). "Trimmed L-moments". Computational Statistics & Data Analysis 43 (3): 299–314. doi:10.1016/S0167-9473(02)00250-5.