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 E {displaystyle mathrm {E} } 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).