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Biometrika 1999 86(2):351-364; doi:10.1093/biomet/86.2.351
© 1999 by Biometrika Trust
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Convex linear combinations of compositions

J AitchisonA1 and J Bacon-ShoneA2

A1 Department of Statistics, University of Glasgow, Glasgow G12 8QQ, UK E-mail: John.Aitchison@btinternet.com A2 Social Sciences Research Centre, University of Hong Kong, Pokfulam Road, Hong Kong, ROC E-mail: johnbs@hku.hk

When a sampled target composition is suspected of being a mixture of different compositions from a number of independent sources the question of the nature of the mixing mechanism arises. For the resolution of this question several models involving convex linear mixtures of compositions are considered and in particular the distributional problem of describing the pattern of variability of the target compositions, given information about the source distributions, is resolved in terms of approximations involving logistic normal and logistic skew normal distributions. The quality of these approximations is shown to be satisfactory through a series of simulations briefly reported. The modelling and subsequent statistical inference are motivated by an illustrative application to investigating the nature of pollution at three fishing locations in a Scottish loch.

Key Words: Dirichlet distribution; End-member problem; Lattice testing of hypotheses; Logistic normal distribution; Mixing of compositions; Multivariate skew normal distribution.


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