| Abstract: | The sample space of compositional data, a simplex, induces a different kind of geometry, known as Aitchison geometry, with
the Euclidean space property. For this reason, the standard statistical analysis is not meaningful here, and this is also
true for measures of location and covariance. The measure of location, called centre, is the best linear unbiased estimator
of the central tendency of the distribution of a random composition with respect to the geometry on the simplex (Pawlowsky-Glahn
and Egozcue in Stoch Envir Res Risk Ass, 15:384–398, 2001; Math Geol, 34:259–274, 2002). Its covariance structure is described
through a variation matrix, which induces the so called total variation as a measure of dispersion. The aim of the paper is
to show that its sample counterpart has theoretical properties, corresponding to the standard multivariate case, like unbiasedness
and convergence in probability. Moreover, its distribution in the case of normality on the simplex is developed. |
