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. For a random variable X, the rth population L-moment is where Xk:n denotes the kth order statistic (kth smallest value) in an independent sample of size n from the distribution of X and denotes expected value. 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, as used in the k-order finite difference (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. 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 n-element sample can be the jth 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 x(i) is the ith 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. 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.