The Skew-Normal Distribution on the Simplex

Abstract

Density functions on the simplex defined with respect to the Lebesgue measure can change from unimodality to multimodality with perturbation. This phenomenon is induced by the incompatibility of the Aitchison geometry and the Lebesgue measure. A Lebesgue-type measure, compatible with the algebraic geometric structure of the simplex, is used here to define the skew-normal density on the simplex as the Radon-Nykodym derivative with respect to it. Similarities and differences between the densities obtained using the different measures are analyzed.

Keywords:

• Aitchison geometry
• Constrained sample space
• Random composition

Mathematics Subject Classification:

• Primary 62M10
• Secondary 62H05

Acknowledgments

The authors thank two anonymous referees for their suggestions which greatly improved the article. This research was supported by the Dirección General de Enseñanza Superior (DGES) of the Spanish Ministry for Education and Culture through the project BFM2003-05640/MATE.