A contribution to multivariate L-moments: L-comoment matrices

Multivariate statistical analysis relies heavily on moment assumptions of second order and higher. With increasing interest in heavy-tailed distributions, however, it is desirable to describe dispersion, skewness, and kurtosis under merely first order moment assumptions. Here, the univariate L-moments of Hosking [L-moments: analysis and estimation of distributions using linear combinations of order statistics, J. Roy. Statist. Soc. Ser. B 52 (1990) 105-124] are extended to "L-comoments" analogous to covariance. For certain models, the second order case yields correlational analysis coherent with classical correlation but also meaningful under just first moment assumptions. We develop properties and estimators for L-comoments, illustrate for several multivariate models, examine behavior of sample multivariate L-moments with heavy-tailed data, and discuss applications to financial risk analysis and regional frequency analysis.