Multivariate Three-Parameter Log-Elliptical Distributions

The three-parameter log-elliptical distribution class is developed for the general situation in which the hypothesis of independence for the elements in a sample is not assumed. The parameter estimators are theoretically showed to be invariant under all distributions in the class by considering only a change in the constant of the scale parameter estimator. An estimation procedure based on the three-parameter lognormal distribution is proposed for the parameter estimation problem in any three-parameter log-elliptical distribution. Two classical lognormal data sets are analyzed without assuming independence in the sample in order to illustrate the proposed estimation procedure.