View source for L-moment

{{Short description|Statistical sequence characterizing probability distributions}}
{{Use American English|date = January 2019}}

In [[statistics]], '''L-moments''' are a sequence of statistics used to summarize the shape of a [[probability distribution]].<ref name="hos:90">{{cite journal | last=Hosking | first=J.R.M. | year=1990 | title=L-moments: analysis and estimation of distributions using linear combinations of order statistics | journal=Journal of the Royal Statistical Society, Series B | volume=52 | issue=1 | pages=105&ndash;124 | doi=10.1111/j.2517-6161.1990.tb01775.x | jstor=2345653}}</ref><ref name="hos:92">{{cite journal | last=Hosking | first= J.R.M. | year=1992 | title=Moments or L moments? An example comparing two measures of distributional shape | journal=The American Statistician | volume=46 | number=3 | pages=186&ndash;189 | jstor=2685210 | doi=10.2307/2685210}}</ref><ref name="hos:96">{{cite journal | last=Hosking | first=J.R.M. | year=2006 | title=On the characterization of distributions by their L-moments | journal=Journal of Statistical Planning and Inference | volume=136 | pages=193&ndash;198 | doi=10.1016/j.jspi.2004.06.004}}</ref><ref>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}}</ref> They are [[linear combination]]s of [[order statistic]]s ([[L-statistic]]s) analogous to conventional [[Moment (mathematics)|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 moment]]s. 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 {{mvar|X}}, the {{mvar|r}}th population L-moment is<ref name=hos:90/>

:<math>
\lambda_r = \frac{\ 1\ }{r} \sum_{k=0}^{r-1} (-1)^k \binom{r-1}{k} \operatorname\mathbb{E} \{\ X_{r-k:r}\ \}\ ,
</math>

where {{mvar| X{{sub|k:n}} }} denotes the {{mvar|k}}th [[order statistic]] ({{mvar|k}}th smallest value) in an [[statistical independence|independent]] [[statistical sample|sample]] of size {{mvar|n}} from the distribution of {{mvar|X}} and <math>\ \mathbb{E}\ </math> denotes [[expected value]] [[operator (mathematics)|operator]]. In particular, the first four population L-moments are

:<math>
\lambda_1 = \operatorname{\mathbb E}\,\! \{\ X\ \}
</math>

:<math>
\lambda_2 = \frac{\ 1\ }{2} \Bigl(\ \operatorname{\mathbb E}\,\! \{\ X_{2:2}\ \} - \operatorname{\mathbb E}\,\! \{\ X_{1:2}\ \}\ \Bigr) 
</math>

:<math>
\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) 
</math>

:<math>
\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) ~.
</math>

Note that the coefficients of the {{mvar|r}}th L-moment are the same as in the {{mvar|r}}th term of the [[binomial transform]], as used in the {{mvar|r}}-order [[finite difference]] (finite analog to the derivative).

The first two of these L-moments have conventional names:
:<math>\  \lambda_1\  </math> is the "mean", "L-mean", or "L-location",
:<math>\  \lambda_2\  </math> is the "L-scale".
The L-scale is equal to half the [[Mean absolute difference]].<ref name=jones:02/>

===Analytic calculation===
Expectations are often defined in terms of [[probability density function]]s, but the connection in terms of these between the order statistics <math> X_{r:n} </math> and their underlying random variable <math> X </math> is rather remote. A closer connection can be found in terms of [[cumulative distribution function]]s (CDFs), since these (see [[Order_statistic#Cumulative_distribution_function_of_order_statistics|this section]]) satisfy
<math display="block">
  F_{X_{r:n}}(x) = \sum_{j=r}^n \binom{n}{j} F_X(x)^j \bigl( 1 - F_X(x) \bigr)^{n-j}
  \text{.}
</math>
In particular one may define polynomials <math> b_{r:n}(y) = \sum_{j=r}^n \binom{n}{j} y^j (1-y)^{n-j} </math> and express <math> F_{X_{r:n}} = b_{r:n} \circ F_X </math>.

Having a CDF <math> F_X </math>, the expectation <math> \mathbb{E}\{X\} </math> may be expressed using a [[Stieltjes integral]] as
<math display="block">
  \mathbb{E}\{X\} = \int_{\mathbb{R}} x \, dF_X(x)
  \text{,}
</math>
thus
<math display="block">
  \mathbb{E}\{X_{r:n}\} = 
  \int_{\mathbb{R}} x \, d(b_{r:n} \circ F_X)(x) =
  \int_{\mathbb{R}} x b_{r:n}'\bigl( F_X(x) \bigr) \, dF_X(x)
</math>
where <math> b_{r:n}' </math> is straight off the derivative of <math> b_{r:n} </math>. This integral can often be made more tractable by introducing the [[quantile function]] <math> Q_X </math> via the [[change of variables]] <math> y = F_X(x), x = Q_X(y) </math>:
<math display="block">
  \mathbb{E}\{X_{r:n}\} = 
  \int_{\mathbb{R}} x b_{r:n}'\bigl( F_X(x) \bigr) \, dF_X(x) =
  \int_0^1 Q_X(y) b_{r:n}'(y) \, dy
  \text{.}
</math>
Since the L-moments are linear combinations of such expectations, the corresponding integrals can be combined into one for each moment, where the integrand is <math> Q_X(y) </math> times a polynomial. We have<ref name="hos:90"/>
<math display="block">
  \lambda_n = \int_0^1 Q_X(y) \widetilde{P}_{n-1}(y) \, dy
</math>
where
<math display="block">
  \widetilde{P}_m(y) = \sum_{k=0}^m (-1)^{m-k} \binom{m}{k} \binom{m+k}{k} y^k
</math>
are the [[shifted Legendre polynomials]], orthogonal on <math> [0,1] </math>.

In particular
<math display="block">
  \lambda_1 = \int_0^1 Q_X(y) \, dy
  \text{,}
</math>
<math display="block">
  \lambda_2 = \int_0^1 Q_X(y) (2y-1) \, dy
  \text{,}
</math>
<math display="block">
  \lambda_3 = \int_0^1 Q_X(y) (6y^2 - 6y + 1) \, dy
  \text{,}
</math>
<math display="block">
  \lambda_4 = \int_0^1 Q_X(y) (20y^3 - 30y^2 + 12y - 1) \, dy
  \text{.}
</math>

=== Sillitto's Theorem ===
The above integral formula for <math> \lambda_n </math> has the form of a [[generalized Fourier series|generalised Fourier coefficient]], and they appeared as such in the literature years before being named moments. In the notation of this article, Sillitto<ref>{{cite journal |last1=Sillitto |first1=G. P. |title=Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample |journal=Biometrika |date=1969 |volume=56 |page=641–650}}</ref> proved

: '''Theorem.''' Let <math> X </math> be a real-valued continuous random variable with finite variance, quantile function <math> Q_X(y) </math> and L-moments <math> \{ \lambda_r \}_{r=1}^\infty </math>. Then the representation <math display="block"> Q_X(y) = \sum_{r=1}^\infty (2r-1) \lambda_r \widetilde{P}_{r-1}(y) \qquad \text{for } 0 < y < 1 </math> is [[Lp space|convergent in <math> L^2 </math> norm]].

However Hosking<ref name="hos:90"/> cautions that [[partial sum]]s of this series tend to give poor approximations for the tails of the distribution, and need not be monotonic. Similar problems arise with the [[Cornish–Fisher expansion]] of <math> Q_X </math> in terms of the [[cumulant]]s of <math> X </math>.

==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 <math>\left\{ x_1 < \cdots < x_j < \cdots < x_r \right\},</math> hence averaging by dividing by the [[binomial coefficient]]:
:<math>
\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 ~.
</math>

Grouping these by order statistic counts the number of ways an element of an {{mvar|n}}&nbsp;element sample can be the {{mvar|j}}th element of an {{mvar|r}}&nbsp;element subset, and yields formulas of the form below. Direct estimators for the first four L-moments in a finite sample of {{mvar|n}}&nbsp;observations are:<ref name=wang:96>
{{cite journal
 |last=Wang |first=Q.J.
 |year=1996
 |title=Direct sample estimators of L-moments
 |journal=Water Resources Research
 |volume=32 |issue=12 |pages=3617–3619
 |doi=10.1029/96WR02675
|bibcode=1996WRR....32.3617W
 }}
</ref>

:<math>\ell_1 = \frac{1}{\ \tbinom{n}{1}\ } \sum_{i=1}^n\ x_{(i)}\ </math>
:<math>\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)}\ </math>
:<math>\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)}\ </math>
:<math>\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)}\ </math>

where {{math|''x''<sub>(''i'')</sub>}} is the {{mvar|i}}th [[order statistic]] and <math>\ \tbinom{\boldsymbol\cdot}{\boldsymbol\cdot}\ </math> is a [[binomial coefficient]]. Sample L-moments can also be defined indirectly in terms of [[probability weighted moment]]s,<ref name=hos:90/><ref name=green:79>
{{cite journal
 |first1=J.A. |last1=Greenwood     |first2=J.M. |last2=Landwehr
 |first3=N.C. |last3=Matalas       |first4=J.R. |last4=Wallis
 |year=1979
 |title=Probability weighted moments: Definition and relation to parameters of several distributions expressed in inverse form
 |journal=Water Resources Research
 |volume=15 |issue=5 |pages=1049–1054
 |doi= 10.1029/WR015i005p01049
 |s2cid=121955257
 |url=https://pdfs.semanticscholar.org/2e01/2d5677fa4edf246d39292ab6e58254e13df2.pdf
 |url-status=dead
 |archive-url=https://web.archive.org/web/20200210192241/https://pdfs.semanticscholar.org/2e01/2d5677fa4edf246d39292ab6e58254e13df2.pdf
 |archive-date=2020-02-10
}}
</ref><ref name=land:79>
{{cite journal
 |first1=J.M. |last1=Landwehr
 |first2=N.C. |last2=Matalas
 |first3=J.R. |last3=Wallis
 |year=1979
 |title=Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles
 |journal=Water Resources Research
 |volume=15 |issue=5 |pages=1055–1064
 |doi=10.1029/WR015i005p01055
|bibcode=1979WRR....15.1055L
 }}
</ref>
which leads to a more efficient [[algorithm]] for their computation.<ref name=wang:96/><ref>
{{cite web
 |title=L&nbsp;moments
 |date=6 January 2006
 |department=NIST Dataplot
 |type=documentation
 |publisher=[[National Institute of Standards and Technology]]
 |website=itl.nist.gov
 |url=http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/lmoment.htm
 |access-date=19 January 2013
}}
</ref>

==L-moment ratios==
A set of ''L-moment ratios'', or scaled L-moments, 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 ''L-skewness'', and <math>\ \tau_4\ ,</math> the ''L-kurtosis''.

L-moment ratios lie within the interval {{nobr|&thinsp;{{math|( −1, 1 )}} .}} Tighter bounds can be found for some specific L-moment ratios; in particular, the L-kurtosis <math>\ \tau_4\ </math> lies in {{nobr|&thinsp;{{math|[ {{sfrac|−| 1 |4}}, 1 ) }} ,}} and
:<math>\ \tfrac{\ 1\ }{4} \left(\ 5\ \tau_3^2 - 1\ \right) \leq \tau_4 < 1 ~.</math><ref name=hos:90/>

A quantity analogous to the [[coefficient of variation]], but based on L-moments, can also be defined:
<math>\ \tau = \lambda_2 / \lambda_1\ ,</math>
which is called the "coefficient of L-variation", or "L-CV". For a non-negative random variable, this lies in the interval {{nobr|&thinsp;{{math| ( 0, 1 ) }}&thinsp;}}<ref name=hos:90/> and is identical to the [[Gini coefficient]].<ref name=rvp:01>
{{cite journal
 | last1=Valbuena  | first1=R.        | last2=Maltamo  | first2=M.
 | last3=Mehtätalo | first3=L.        | last4=Packalen | first4=P.
 | year=2017
 | title=Key structural features of Boreal forests may be detected directly using L-moments from airborne lidar data
 | journal=Remote Sensing of Environment
 | volume=194 | pages=437–446
 | doi=10.1016/j.rse.2016.10.024
 | bibcode=2017RSEnv.194..437V | url=http://oa.upm.es/49935/
}}
</ref>

==Related quantities==

L-moments are statistical quantities that are derived from probability weighted moments<ref>{{cite book|url=https://books.google.com/books?id=gurAnfB4nvUC|pages=3|first1=JRM |last1=Hosking|first2= JR|last2= Wallis|title= Regional Frequency Analysis: An Approach Based on L-moments|year=2005|isbn=978-0521019408|access-date = 22 January 2013|publisher=Cambridge University Press}}</ref>  (PWM) which were defined earlier (1979).<ref name=green:79/> PWM are used to efficiently estimate the parameters of distributions expressable in inverse form such as the [[Gumbel distribution|Gumbel]],<ref name=land:79/> the [[Tukey lambda distribution|Tukey lambda]], and the [[Wakeby distribution|Wakeby]] distributions.

==Usage==
There are two common ways that L-moments 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 (statistics)|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 statistics|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]]&nbsp;(EVT). This application shows the limited robustness of L-moments, i.e. L-statistics are not [[resistant statistic]]s, 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 (statistics)|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.<ref name=hos:90/>

Some appearances of L-moments in the statistical literature include the book by David & Nagaraja (2003, Section 9.9)<ref>{{cite book |last1=David |first1=H. A. |last2=Nagaraja |first2=H. N. |year=2003 |title=Order Statistics |edition=3rd |publisher=Wiley |isbn=978-0-471-38926-2 }}</ref> and a number of papers.<ref name=rvp:01/><ref>{{cite journal |last1=Serfling |first1=R. |last2=Xiao |first2=P. |year=2007 |title=A contribution to multivariate L-moments: L-comoment matrices |journal=Journal of Multivariate Analysis |volume=98 |issue=9 |pages=1765–1781 |doi=10.1016/j.jmva.2007.01.008 |citeseerx=10.1.1.62.4288 }}</ref><ref>{{cite journal |last1=Delicado |first1=P. |last2=Goria |first2=M. N. |year=2008 |title=A small sample comparison of maximum likelihood, moments and L-moments methods for the asymmetric exponential power distribution |journal=Computational Statistics & Data Analysis |volume=52 |issue=3 |pages=1661–1673 |doi=10.1016/j.csda.2007.05.021 }}</ref><ref>{{cite journal |last1=Alkasasbeh |first1=M. R. |last2=Raqab |first2=M. Z. |year=2009 |title=Estimation of the generalized logistic distribution parameters: comparative study |journal=Statistical Methodology |volume=6 |issue=3 |pages=262–279 |doi=10.1016/j.stamet.2008.10.001 }}</ref><ref>{{cite journal |last=Jones |first=M. C. |year=2004 |title=On some expressions for variance, covariance, skewness and L-moments |journal=Journal of Statistical Planning and Inference |volume=126 |issue=1 |pages=97–106 |doi=10.1016/j.jspi.2003.09.001 }}</ref><ref>{{cite journal |last=Jones |first=M. C. |year=2009 |title=Kumaraswamy's distribution: A beta-type distribution with some tractability advantages |journal=Statistical Methodology |volume=6 |issue=1 |pages=70–81 |doi=10.1016/j.stamet.2008.04.001 }}</ref> A number of favourable comparisons of L-moments with ordinary moments have been reported.<ref>{{cite journal |last=Royston |first=P. |year=1992 |title=Which measures of skewness and kurtosis are best? |journal=[[Statistics in Medicine (journal)|Statistics in Medicine]] |volume=11 |issue=3 |pages=333–343 |doi=10.1002/sim.4780110306|pmid=1609174 }}</ref><ref>{{cite journal |last1=Ulrych |first1=T. J. |last2=Velis |first2=D. R. |last3=Woodbury |first3=A. D. |last4=Sacchi |first4=M. D. |year=2000 |title=L-moments and C-moments |journal=Stochastic Environmental Research and Risk Assessment |volume=14 |issue=1 |pages=50–68 |doi=10.1007/s004770050004 |bibcode=2000SERRA..14...50U |s2cid=120542594 }}</ref>

==Values for some common distributions==
The table below gives expressions for the first two L&nbsp;moments and numerical values of the first two L-moment ratios of some common [[continuous probability distribution]]s with constant L-moment ratios.<ref name=hos:90/><ref name=jones:02>
{{cite journal
 |last=Jones |first=M.C.
 |title=Student's simplest distribution
 |journal=[[Journal of the Royal Statistical Society, Series D]]
 |year=2002
 |volume=51 |issue=1 |pages=41–49
 |jstor=3650389 |doi=10.1111/1467-9884.00297
}}
</ref>
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 distribution|log-normal]], [[Gamma distribution|Gamma]], [[generalized Pareto distribution|generalized Pareto]], [[generalized extreme value distribution|generalized extreme value]], and [[generalized logistic distribution|generalized logistic]] distributions.<ref name=hos:90/>

:{| class="wikitable sortable" style="text-align:center;"
|-
! Distribution 
! class="unsortable" | Parameters
! class="unsortable" | mean, {{math|''λ''{{sub|1}} }}
! class="unsortable" | L-scale, {{math|''λ''{{sub|2}} }}
! L-skewness, {{math|''τ''<sub>3</sub>}} 
! L-kurtosis, {{math|''τ''<sub>4</sub>}}
|-
! [[Continuous uniform distribution|Uniform]] 
| {{mvar|a}}, {{mvar|b}} || {{math| {{sfrac| 1 |2}}(''a'' + ''b'') }} || {{math| {{sfrac| 1 |6}}(''b'' – ''a'') }} 
|align="right"| {{math| 0 }}
|align="right"| {{math| 0 }}
|-
! [[Logistic distribution|Logistic]] 
| {{mvar|μ}}, {{mvar|s}} || {{mvar|μ}} || {{mvar|s}} 
|align="right"| 0 
|align="right"| {{sort|0.1667| {{math| {{sfrac| 1 |6}} {{=}} 0.1667 }} }}
|-
! [[Normal distribution|Normal]] 
| {{mvar|μ}}, {{math| ''σ''{{sup|2}} }} || {{mvar|μ}} || {{math| {{sfrac| σ | {{sqrt|π}} }} }}
|align="right"| 0 
|align="right"| {{sort|0.1226| {{math| 30 {{sfrac| ''[[Magic angle|{{mvar|θ}}{{sub|m}}]]'' |π }} - 9 {{=}} 0.1226 }} }} 
|-
! [[Laplace distribution|Laplace]] 
| {{mvar|μ}}, {{mvar|b}} || {{mvar|μ}} || {{math| {{sfrac| 3 | 4 }} ''b'' }}
|align="right"| 0 
|align="right"| {{sort|0.2357| {{math| {{sfrac| 1 | 3 {{sqrt|2}} }} {{=}} 0.2357 }} }}
|-
! [[Student's t-distribution|Student's ''t'']], 2 [[degrees of freedom (statistics)|d.f.]]
| {{math| ''ν'' {{=}} 2 }} || 0 || {{math| {{sfrac| ''π'' | 2 {{sqrt|2}} }} {{=}} 1.111  }}
|align="right"| 0 
|align="right"| {{sort|0.375| {{math| {{sfrac| 3 | 8 }} {{=}} 0.375 }} }}
|-
! [[Student's t-distribution|Student's ''t'']], 4 [[degrees of freedom (statistics)|d.f.]]
| {{math| ''ν'' {{=}} 4 }} || 0 || {{math| {{sfrac| 15 |64}} ''π'' {{=}} 0.7363 }}
|align="right"| 0 
|align="right"| {{sort|0.2168| {{math| {{sfrac| 111 |512}} {{=}} 0.2168 }} }} 
|-
! [[Exponential distribution|Exponential]] 
| {{mvar|λ}} || {{math| {{sfrac| 1 | ''λ'' }} }} || {{math| {{sfrac| 1 | 2 ''λ'' }} }}
|align="right"| {{sort|0.3333| {{math| {{sfrac| 1 |3}} {{=}} 0.3333 }} }}
|align="right"| {{sort|0.1667| {{math| {{sfrac| 1 |6}} {{=}} 0.1667 }} }}
|-
! [[Gumbel distribution|Gumbel]] 
| {{mvar|μ}}, {{mvar|β}}  || {{mvar|μ}} + ''[[Euler–Mascheroni constant|{{mvar|γ}}{{sub|e}}]] {{mvar|β}} || {{nobr| {{math| ''β'' log{{sub|2}}(3) }} }}
|align="right"| {{sort|0.1699| {{math| 2 log{{sub|2}}(3) - 3 {{=}} 0.1699  }} }}
|align="right"| {{sort|0.1504| {{math| 16 - 10 log{{sub|2}}(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]], {{mvar|γ}}{{sub|e}} is the {{nobr| [[Euler–Mascheroni constant]] {{math| 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]].<ref>{{cite journal|last=Elamir|first=Elsayed A. H.|author2=Seheult, Allan H. |title=Trimmed L-moments|journal=Computational Statistics & Data Analysis|year=2003|volume=43|issue=3|pages=299–314|doi=10.1016/S0167-9473(02)00250-5}}</ref>

==See also==
*[[L-estimator]]

==References==
{{reflist}}

==External links==
* [http://www.research.ibm.com/people/h/hosking/lmoments.html The L-moments page] Jonathan R.M. Hosking, [[IBM Research]]
* [http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/lmoment.htm L Moments.] [[Dataplot]] reference manual, vol. 1, auxiliary chapter. [[National Institute of Standards and Technology]], 2006. Accessed 2010-05-25.
* [https://github.com/jorenham/lmo Lmo] lightweight Python includes functions for fast calculation of L-moments, trimmed L-moments, and multivariate L-comoments.

{{theory of probability distributions|state=expanded}}
{{Statistics|descriptive}}

[[Category:Moment (mathematics)]]
[[Category:Summary statistics]]