Some Problems with the Distributions of Compositional Data

The famous book "The Statistical Analysis of Compositonal Data" written by J.Ai-tehison （1986）, was translated into Chinese and published in 1989. Many statistical methods were accepted by applied statisticans. In this paper, we discuss three problems with another view which is different to the book.
1) The Dirichlet distribution and the independence of the components of the basis vector. We show in the paper that there are many basis vectors, with dependent component variables, the compositional vector is distributed with Dirichlet distribution.
2) The additive logistical normal distribution . Most parts of the book 1 was discussed the statstical analysis of the logistical normal distribution （additive and multiplicative ）. It is not necessary to deduce the multiplicative logistical normal distribution for discussing the independence of sub-compositional vectors.In this paper we give a theorem about the independence of sub-composition under the additive logistical normal distribution.
3) Independence of the composition vector and the total amount. In this paper, we emphasize that the independence of the composition vector and the total amount is very important, otherwise, it is not neccesary to discuss the stastistical analysis of compositional vectors. We can derive the distribution family of composition from the independence property.