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Statistical properties of the total variation estimator for compositional data. (English) Zbl 1434.62093
Summary: 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 [V. Pawlowsky-Glah and J. J. Egozcue, Stoch. Environ. Res. Risk Assess. 15, No. 5, 384–398 (2001; Zbl 0987.62001); Math. Geol. 34, No. 3, 259–274 (2002; Zbl 1031.86007)]. 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.

MSC:
62H11 Directional data; spatial statistics
62E20 Asymptotic distribution theory in statistics
62H10 Multivariate distribution of statistics
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