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 $r$ th population L-moment is^[1]

\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 $k$ th order statistic ( $k$ th smallest value) in an independent sample of size $n$ from the distribution of $X$ and $\ \mathbb {E} \$ denotes expected value operator. In particular, the first four population L-moments are

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

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

\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 )}

\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 $r$ th L-moment are the same as in the $r$ th 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:

\ \lambda _{1}\

is the "mean", "L-mean", or "L-location",

\ \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 $\left\{x_{1}<\cdots <x_{j}<\cdots <x_{r}\right\},$ hence averaging by dividing by the binomial coefficient:

\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 $j$ th 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]

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

\ell _{2}={\frac {1}{\ 2\cdot {\tbinom {n}{2}}\ }}\sum _{i=1}^{n}\ {\Bigl }\ x_{(i)}\

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