Exact variance structure of sample L-moments

Abstract

Population L-moments have been proposed as alternatives to central moments for describing distribution location, dispersion and shape, and their sample estimates are unbiased. However, only asymptotic variances and covariances of their estimates have been reported. In this article, we derive expressions for exact variances and covariances of sample L-moments for any sample size n in terms of first- and second-order moments of order statistics from small sample sizes which do not depend on n. Various applications of these result are discussed. We also derive distribution-free unbiased estimators of the variances and covariances of sample L-moments, and report the results of a simulation study to investigate and compare the sampling distributions of standardised L-moments using exact, asymptotic and estimated standard errors. In particular, a new test of symmetry is investigated. Also, approximate standard errors of ratios of sample L-moments, used to estimate ratios of population L-moments analogous to classical scaled measures of skewness and kurtosis, are exemplified.