In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, L-moments are a sequence of statistics used to summarize the shape of a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.
They are
linear combinations of
order statistic
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.
Import ...
s (
L-statistics) analogous to conventional
moments, and can be used to calculate quantities analogous to
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
,
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal d ...
and
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
). Standardised L-moments are called L-moment ratios and are analogous to
standardized moment
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. ...
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 ''X'', the ''r''th population L-moment is
[
:
where ''X''''k:n'' denotes the ''k''th ]order statistic
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.
Import ...
(''k''th smallest value) in an independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of s ...
of size ''n'' from the distribution of ''X'' and denotes expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
. In particular, the first four population L-moments are
:
:
:
:
Note that the coefficients of the ''k''-th L-moment are the same as in the ''k''-th term of the binomial transform In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to th ...
, as used in the ''k''-order finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
(finite analog to the derivative).
The first two of these L-moments have conventional names:
:
:
The L-scale is equal to half the Mean absolute difference.[
]
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 hence averaging by dividing by the binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
:
:
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:
:
:
:
:
where is the th order statistic
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.
Import ...
and is a binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. Sample L-moments can also be defined indirectly in terms of probability weighted moments, which leads to a more efficient algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
for their computation.[
]
L-moment ratios
A set of ''L-moment ratios'', or scaled L-moments, is defined by
:
The most useful of these are , called the ''L-skewness'', and , 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 lies in \tfrac(5\tau_3^2-1) \leq \tau_4 < 1.[
A quantity analogous to the coefficient of variation">¼,1), and
:][
A quantity analogous to the coefficient of variation], but based on L-moments, can also be defined:
which is called the "coefficient of L-variation", or "L-CV". For a non-negative random variable, this lies in the interval (0,1)[ and is identical to the Gini coefficient.]
Related quantities
L-moments are statistical quantities that are derived from probability weighted moments (PWM) which were defined earlier (1979).[ PWM are used to efficiently estimate the parameters of distributions expressable in inverse form such as the Gumbel,][ the Tukey, and the Wakeby distributions.
]
Usage
There are two common ways that L-moments are used, in both cases analogously to the conventional moments:
# As summary statistics
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in
* a measure of ...
for data.
# To derive estimators for the parameters of probability distributions
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, 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
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
methods; however using L-moments provides a number of advantages. Specifically, L-moments are more robust
Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
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
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
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
Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the pr ...
(EVT). This application shows the limited robustness of L-moments, i.e. L-statistics are not resistant statistic
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such ...
s, as a single extreme value can throw them off, but because they are only linear (not higher-order statistics
In statistics, the term higher-order statistics (HOS) refers to functions which use the third or higher power of a sample, as opposed to more conventional techniques of lower-order statistics, which use constant, linear, and quadratic terms (zer ...
), 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
In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situa ...
with low degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
). A finite variance is required in addition in order for the standard errors of estimates of the L-moments to be finite.[
Some appearances of L-moments in the statistical literature include the book by David & Nagaraja (2003, Section 9.9) and a number of papers.][ A number of favourable comparisons of L-moments with ordinary moments have been reported.
]
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 distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s with constant L-moment ratios.
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
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
, Gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
, generalized Pareto, generalized extreme value, and generalized logistic distributions.[
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
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
0.57721... .
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
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
.
See also
*L-estimator
In statistics, an L-estimator is an estimator which is a linear combination of order statistics of the measurements (which is also called an L-statistic). This can be as little as a single point, as in the median (of an odd number of values), or as ...
References
External links
The L-moments page
Jonathan R.M. Hosking, IBM Research
IBM Research is the research and development division for IBM, an American multinational information technology company headquartered in Armonk, New York, with operations in over 170 countries. IBM Research is the largest industrial research org ...
L Moments.
Dataplot reference manual, vol. 1, auxiliary chapter. National Institute of Standards and Technology
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
, 2006. Accessed 2010-05-25.
{{Statistics, descriptive
Moment (mathematics)
Summary statistics