Frequent Links
Lmoment
In statistics, Lmoments 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 (Lstatistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the Lscale, Lskewness and Lkurtosis respectively (the Lmean is identical to the conventional mean). Standardised Lmoments are called Lmoment ratios and are analogous to standardized moments. Just as for conventional moments, a theoretical distribution has a set of population Lmoments. Sample Lmoments can be defined for a sample from the population, and can be used as estimators of the population Lmoments.
Contents
Population Lmoments
For a random variable X, the rth population Lmoment is^{[1]}
 <math>
\lambda_r = r^{1} \sum_{k=0}^{r1} {(1)^k \binom{r1}{k} \mathrm{E}X_{rk:r}}, </math>
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 <math>\mathrm{E}</math> denotes expected value. In particular, the first four population Lmoments are
 <math>
\lambda_1 = \mathrm{E}X </math>
 <math>
\lambda_2 = (\mathrm{E}X_{2:2}  \mathrm{E}X_{1:2})/2 </math>
 <math>
\lambda_3 = (\mathrm{E}X_{3:3}  2\mathrm{E}X_{2:3} + \mathrm{E}X_{1:3})/3 </math>
 <math>
\lambda_4 = (\mathrm{E}X_{4:4}  3\mathrm{E}X_{3:4} + 3\mathrm{E}X_{2:4}  \mathrm{E}X_{1:4})/4. </math>
Note that the coefficients of the kth Lmoment are the same as in the kth term of the binomial transform, as used in the korder finite difference (finite analog to the derivative).
The first two of these Lmoments have conventional names:
 <math>\lambda_1 = \text{mean, Lmean or Llocation},</math>
 <math>\lambda_2 = \text{Lscale}.</math>
The Lscale is equal to half the mean difference.^{[5]}
Sample Lmoments
The sample Lmoments can be computed as the population Lmoments of the sample, summing over relement subsets of the sample <math>\left\{ x_1 < \cdots < x_j < \cdots < x_r \right\},</math> hence averaging by dividing by the binomial coefficient:
 <math>
\lambda_r = r^{1}{\tbinom{n}{r}}^{1} \sum_{x_1 < \cdots < x_j < \cdots < x_r} {(1)^{rj} \binom{r1}{j} x_j}. </math>
Grouping these by order statistic counts the number of ways an element of an nelement sample can be the jth element of an relement subset, and yields formulas of the form below. Direct estimators for the first four Lmoments in a finite sample of n observations are:^{[6]}
 <math>\ell_1 = {\tbinom{n}{1}}^{1} \sum_{i=1}^n x_{(i)}</math>
 <math>\ell_2 = \tfrac{1}{2} {\tbinom{n}{2}}^{1} \sum_{i=1}^n \left\{ \tbinom{i1}{1}  \tbinom{ni}{1} \right\} x_{(i)}</math>
 <math>\ell_3 = \tfrac{1}{3} {\tbinom{n}{3}}^{1} \sum_{i=1}^n \left\{ \tbinom{i1}{2}  2\tbinom{i1}{1}\tbinom{ni}{1} + \tbinom{ni}{2} \right\} x_{(i)}</math>
 <math>\ell_4 = \tfrac{1}{4} {\tbinom{n}{4}}^{1} \sum_{i=1}^n \left\{ \tbinom{i1}{3}  3\tbinom{i1}{2}\tbinom{ni}{1} + 3\tbinom{i1}{1}\tbinom{ni}{2}  \tbinom{ni}{3} \right\} x_{(i)}</math>
where x_{(i)} is the ith order statistic and <math>\tbinom{\cdot}{\cdot}</math> is a binomial coefficient. Sample Lmoments 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]}
Lmoment ratios
A set of Lmoment ratios, or scaled Lmoments, is defined by
 <math>\tau_r = \lambda_r / \lambda_2, \qquad r=3,4, \dots. </math>
The most useful of these are <math>\tau_3</math>, called the Lskewness, and <math>\tau_4</math>, the Lkurtosis.
Lmoment ratios lie within the interval (–1, 1). Tighter bounds can be found for some specific Lmoment ratios; in particular, the Lkurtosis <math>\tau_4</math> lies in [¼,1), and
 <math>\tfrac{1}{4}(5\tau_3^21) \leq \tau_4 < 1.</math>^{[1]}
A quantity analogous to the coefficient of variation, but based on Lmoments, can also be defined: <math>\tau = \lambda_2 / \lambda_1, </math> which is called the "coefficient of Lvariation", or "LCV". For a nonnegative random variable, this lies in the interval (0,1)^{[1]} and is identical to the Gini coefficient.
Related quantities
Lmoments are statistical quantities that are derived from probability weighted moments^{[10]} (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, and the Wakeby distributions.
Usage
There are two common ways that Lmoments are used, in both cases analogously to the conventional moments:
 As summary statistics for data.
 To derive estimators for the parameters of probability distributions, applying the method of moments to the Lmoments 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 Lmoments provides a number of advantages. Specifically, Lmoments are more robust than conventional moments, and existence of higher Lmoments only requires that the random variable have finite mean. One disadvantage of Lmoment 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 Lmoment ratio than e.g. the studentt 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 Lscale is taken it will be far less sensitive to this data value. Consequently Lmoments 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 Lmoments. One example of this is using Lmoments as summary statistics in extreme value theory (EVT). This application shows the limited robustness of Lmoments, i.e. Lstatistics are not resistant statistics, as a single extreme value can throw them off, but because they are only linear (not higherorder statistics), they are less affected by extreme values than conventional moments.
Another advantage Lmoments have over conventional moments is that their existence only requires the random variable to have finite mean, so the Lmoments 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 Lmoments to be finite.^{[1]}
Some appearances of Lmoments in the statistical literature include the book by David & Nagaraja (2003, Section 9.9)^{[11]} and a number of papers.^{[12]}^{[13]}^{[14]}^{[15]}^{[16]} A number of favourable comparisons of Lmoments with ordinary moments have been reported.^{[17]}^{[18]}
Values for some common distributions
The table below gives expressions for the first two Lmoments and numerical values of the first two Lmoment ratios of some common continuous probability distributions with constant Lmoment ratios.^{[1]}^{[5]} More complex expressions have been derived for some further distributions for which the Lmoment ratios vary with one or more of the distributional parameters, including the lognormal, Gamma, generalized Pareto, generalized extreme value, and generalized logistic distributions.^{[1]}
Distribution  Parameters  mean, λ_{1}  Lscale, λ_{2}  Lskewness, τ_{3}  Lkurtosis, τ_{4} 

Uniform  a, b  (a+b) / 2  (b–a) / 6  0  0 
Logistic  μ, s  μ  s  0  ^{1}⁄_{6} = 0.1667 
Normal  μ, σ^{2}  μ  σ / √π  0  0.1226 
Laplace  μ, b  μ  3b / 4  0  1 / (3√2) = 0.2357 
Student's t, 2 d.f.  ν = 2  0  π/2^{3/2} = 1.111  0  ^{3}⁄_{8} = 0.375 
Student's t, 4 d.f.  ν = 4  0  15π/64 = 0.7363  0  111/512 = 0.2168 
Exponential  λ  1 / λ  1 / (2λ)  ^{1}⁄_{3} = 0.3333  ^{1}⁄_{6} = 0.1667 
Gumbel  μ, β  μ + γβ  β log 2  0.1699  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, γ is the Euler–Mascheroni constant 0.57721… .
Extensions
Trimmed Lmoments are generalizations of Lmoments that give zero weight to extreme observations. They are therefore more robust to the presence of outliers, and unlike Lmoments they may be welldefined for distributions for which the mean does not exist, such as the Cauchy distribution.^{[19]}
See also
References
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Hosking, J.R.M. (1990). "Lmoments: analysis and estimation of distributions using linear combinations of order statistics". Journal of the Royal Statistical Society, Series B 52: 105–124. JSTOR 2345653.
 ^ Hosking, J.R.M. (1992). "Moments or L moments? An example comparing two measures of distributional shape". The American Statistician 46 (3): 186–189. JSTOR 2685210.
 ^ Hosking, J.R.M. (2006). "On the characterization of distributions by their Lmoments". Journal of Statistical Planning and Inference 136: 193–198. doi:10.1016/j.jspi.2004.06.004.
 ^ Asquith, W.H. (2011) Distributional analysis with Lmoment statistics using the R environment for statistical computing, Create Space Independent Publishing Platform, [printondemand], ISBN 1463508417
 ^ ^{a} ^{b} Jones, M.C. (2002). "Student's Simplest Distribution". Journal of the Royal Statistical Society, Series D 51 (1): 41–49. JSTOR 3650389. doi:10.1111/14679884.00297.
 ^ ^{a} ^{b} Wang, Q. J. (1996). "Direct Sample Estimators of L Moments". Water Resources Research 32 (12): 3617–3619. doi:10.1029/96WR02675.
 ^ ^{a} ^{b} Greenwood, JA; Landwehr, JM; Matalas, NC; Wallis, JR (1979). "Probability Weighted Moments: Definition and relation to parameters of several distributions expressed in inverse form". Water Resources Research 15: 1049–1054. doi:10.1029/WR015i005p01049. Retrieved 17 January 2013.
 ^ ^{a} ^{b} Landwehr, JM; Matalas, NC; Wallis, JR (1979). "Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles". Water Resources Research 15: 1055–1064. doi:10.1029/WR015i005p01055. Retrieved 4 February 2013.
 ^ L Moments, 6 January 2006, retrieved 19 January 2013 NIST Dataplot documentation
 ^ Hosking, JRM; Wallis, JR (2005). Regional Frequency Analysis: An Approach Based on Lmoments. Cambridge University Press. p. 3. ISBN 0521019400. Retrieved 22 January 2013.
 ^ David, H. A.; Nagaraja, H. N. (2003). Order Statistics (3rd ed.). Wiley. ISBN 0471389269.
 ^ Serfling, R.; Xiao, P. (2007). "A contribution to multivariate Lmoments: Lcomoment matrices". Journal of Multivariate Analysis 98 (9): 1765–1781. doi:10.1016/j.jmva.2007.01.008.
 ^ Delicado, P.; Goria, M. N. (2008). "A small sample comparison of maximum likelihood, moments and Lmoments methods for the asymmetric exponential power distribution". Computational Statistics & Data Analysis 52 (3): 1661–1673. doi:10.1016/j.csda.2007.05.021.
 ^ 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.
 ^ Jones, M. C. (2004). "On some expressions for variance, covariance, skewness and Lmoments". Journal of Statistical Planning and Inference 126 (1): 97–106. doi:10.1016/j.jspi.2003.09.001.
 ^ Jones, M. C. (2009). "Kumaraswamy's distribution: A betatype distribution with some tractability advantages". Statistical Methodology 6 (1): 70–81. doi:10.1016/j.stamet.2008.04.001.
 ^ Royston, P. (1992). "Which measures of skewness and kurtosis are best?". Statistics in Medicine 11 (3): 333–343. doi:10.1002/sim.4780110306.
 ^ Ulrych, T. J.; Velis, D. R.; Woodbury, A. D.; Sacchi, M. D. (2000). "Lmoments and Cmoments". Stochastic Environmental Research and Risk Assessment 14 (1): 50–68. doi:10.1007/s004770050004.
 ^ Elamir, Elsayed A. H.; Seheult, Allan H. (2003). "Trimmed Lmoments". Computational Statistics & Data Analysis 43 (3): 299–314. doi:10.1016/S01679473(02)002505.
External links
 The Lmoments page Jonathan R.M. Hosking, IBM Research
 L Moments. Dataplot reference manual, vol. 1, auxiliary chapter. National Institute of Standards and Technology, 2006. Accessed 20100525.
