In statistics, Lmoments^{[1]}^{[2]}^{[3]} are a sequence of statistics used to summarize the shape of a probability distribution. They are Lstatistics (linear combinations of order statistics, hence the "L" for "linear") 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.
Population Lmoments
For a random variable X, the rth population Lmoment is^{[1]}
 $$
\lambda_r = r^{1} \sum_{k=0}^{r1} {(1)^k \binom{r1}{k} \mathrm{E}X_{rk:r}},
where X_{k:n} denotes the k^{th} order statistic (k^{th} smallest value) in an independent sample of size n from the distribution of X and $\backslash mathrm\{E\}$ denotes expected value. In particular, the first four population Lmoments are
 $$
\lambda_1 = \mathrm{E}X
 $$
\lambda_2 = (\mathrm{E}X_{2:2}  \mathrm{E}X_{1:2})/2
 $$
\lambda_3 = (\mathrm{E}X_{3:3}  2\mathrm{E}X_{2:3} + \mathrm{E}X_{1:3})/3
 $$
\lambda_4 = (\mathrm{E}X_{4:4}  3\mathrm{E}X_{3:4} + 3\mathrm{E}X_{2:4}  \mathrm{E}X_{1:4})/4.
Note that the coefficients of the kth Lmoment are the same as in the kth term of the binomial transform, as used in the korder finite difference (finite analog to the derivative).
The first two of these Lmoments have conventional names:
 $\backslash lambda\_1\; =\; \backslash text\{mean,\; Lmean\; or\; Llocation\},$
 $\backslash lambda\_2\; =\; \backslash text\{Lscale\}.$
The Lscale is equal to half the mean difference.^{[4]}
Sample Lmoments
The sample Lmoments can be computed as the population Lmoments of the sample, summing over relement subsets of the sample $\backslash left\backslash \{\; x\_1\; <\; \backslash cdots\; <\; x\_j\; <\; \backslash cdots\; <\; x\_r\; \backslash right\backslash \},$ hence averaging by dividing by the binomial coefficient:
 $$
\lambda_r = r^{1}{\tbinom{n}{r}}^{1} \sum_{x_1 < \cdots < x_j < \cdots < x_r} {(1)^{rj} \binom{r1}{j} x_j}.
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:^{[5]}
 $\backslash ell\_1\; =\; \{\backslash tbinom\{n\}\{1\}\}^\{1\}\; \backslash sum\_\{i=1\}^n\; x\_\{(i)\}$
 $\backslash ell\_2\; =\; \backslash tfrac\{1\}\{2\}\; \{\backslash tbinom\{n\}\{2\}\}^\{1\}\; \backslash sum\_\{i=1\}^n\; \backslash left\backslash \{\; \backslash tbinom\{i1\}\{1\}\; \; \backslash tbinom\{ni\}\{1\}\; \backslash right\backslash \}\; x\_\{(i)\}$
 $\backslash ell\_3\; =\; \backslash tfrac\{1\}\{3\}\; \{\backslash tbinom\{n\}\{3\}\}^\{1\}\; \backslash sum\_\{i=1\}^n\; \backslash left\backslash \{\; \backslash tbinom\{i1\}\{2\}\; \; 2\backslash tbinom\{i1\}\{1\}\backslash tbinom\{ni\}\{1\}\; +\; \backslash tbinom\{ni\}\{2\}\; \backslash right\backslash \}\; x\_\{(i)\}$
 $\backslash ell\_4\; =\; \backslash tfrac\{1\}\{4\}\; \{\backslash tbinom\{n\}\{4\}\}^\{1\}\; \backslash sum\_\{i=1\}^n\; \backslash left\backslash \{\; \backslash tbinom\{i1\}\{3\}\; \; 3\backslash tbinom\{i1\}\{2\}\backslash tbinom\{ni\}\{1\}\; +\; 3\backslash tbinom\{i1\}\{1\}\backslash tbinom\{ni\}\{2\}\; \; \backslash tbinom\{ni\}\{3\}\; \backslash right\backslash \}\; x\_\{(i)\}$
where x_{(i)} is the ith order statistic and $\backslash tbinom\{\backslash cdot\}\{\backslash cdot\}$ is a binomial coefficient. Sample Lmoments can also be defined indirectly in terms of probability weighted moments,^{[1]}^{[6]}^{[7]} which leads to a more efficient algorithm for their computation.^{[5]}^{[8]}
Lmoment ratios
A set of Lmoment ratios, or scaled Lmoments, is defined by
 $\backslash tau\_r\; =\; \backslash lambda\_r\; /\; \backslash lambda\_2,\; \backslash qquad\; r=3,4,\; \backslash dots.$
The most useful of these are $\backslash tau\_3$, called the Lskewness, and $\backslash tau\_4$, the Lkurtosis.
Lmoment ratios lie within the interval (–1, 1). Tighter bounds can be found for some specific Lmoment ratios; in particular, the Lkurtosis $\backslash tau\_4$ lies in [¼,1), and
 $\backslash tfrac\{1\}\{4\}(5\backslash tau\_3^21)\; \backslash leq\; \backslash tau\_4\; <\; 1.$^{[1]}
A quantity analogous to the coefficient of variation, but based on Lmoments, can also be defined:
$\backslash tau\; =\; \backslash lambda\_2\; /\; \backslash lambda\_1,$
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^{[9]} (PWM) which were defined earlier (1979).^{[6]} PWM are used to efficiently estimate the parameters of distributions expressable in inverse form such as the Gumbel,^{[7]} 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 fat 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 just only 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)^{[10]} and a number of papers.^{[11]}^{[12]}^{[13]}^{[14]}^{[15]} A number of favourable comparisons of Lmoments with ordinary moments have been reported.^{[16]}^{[17]}
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]}^{[4]}
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

0.1667 !⅙ = 0.1667

Normal

μ, σ^{2} 
μ 
σ / √π

0

0.1226

Laplace

μ, b 
μ 
3b / 4

0

0.2357 !1 / (3√2) = 0.2357

Student's t, 2 d.f.

ν = 2 
0 
π/2^{3/2} = 1.111

0

0.375 !⅜ = 0.375

Student's t, 4 d.f.

ν = 4 
0 
15π/64 = 0.7363

0

0.2168 !111/512 = 0.2168

Exponential

λ 
1 / λ 
1 / (2λ)

0.3333 !⅓ = 0.3333

0.1667 !⅙ = 0.1667

Gumbel

μ, β 
μ + γ'β 
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.^{[18]}
See also
References
External links
 IBM Research
 National Institute of Standards and Technology, 2006. Accessed 20100525.
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