Analysis and Estimation of Distributions using Linear Combinations of Order Statistics

Abstract

It is well known that the computation of higher order statistics, like skewness and kurtosis (which we call C-moments) is very dependent on sample size and is highly susceptible to the presence of outliers.  To overcome these difficulties, Hosking (1990) has introduced related statistics called L-moments.   L-moments are expectations of certain linear combinations of order statistics.  They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions and hypothetical tests for probability distributions.  The theory of L-moments parallels the theory of conventional moments as this list of applications might suggest.  The main advantage of L-moments over conventional moments is that L-moments, being linear functions of data, suffer less from effects of sampling variability and the probability density functions that are estimated from L-moments are superior estimates to those obtained from conventional moments (maximum likelihood estimates).  L-moments are more robust than conventional moments to outliers in the data and enable more inferences to be made from small samples about an underlying probability distributions.  L-moment derived distributions for real data examples appear to be more consistent sample to sample than pdf’s determined by conventional means.