L-moment
In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution. 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 , the th population L-moment is where denotes the th order statistic (th smallest value) in an independent sample of size from the distribution of and denotes expected value operator. In particular, the first four population L-moments are Note that the coefficients of the th L-moment are the same as in the th term of the binomial transform, as used in the -order finite difference (finite analog to the derivative). The first two of these L-moments have conventional names: is the "mean", "L-mean", or "L-location", is the "L-scale". 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: Grouping these by order statistic counts the number of ways an element of an element sample can be the th element of an element subset, and yields formulas of the form below. Direct estimators for the first four L-moments in a finite sample of observations are: where is the th order statistic and is a binomial coefficient. Sample L-moments can also be defined indirectly in terms of probability weighted moments, which leads to a more efficient algorithm 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 Tighter bounds can be found for some specific L-moment ratios; in particular, the L-kurtosis lies in 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 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 for data. To derive estimators for the parameters of probability distributions, applying the method of moments to the L-moments rather than conventional moments. In addition to doing