成分向量分布的几个问题

Some Problems with the Distributions of Compositional Data

成分统计分析的名著[1]中提出了一些看法,本文就其中三点提出了不同的意见,并给出了实例。(1)关于狄氏分布与独立性;(2)关于乘法逻辑分布的引入有无必要;(3)关于成分与总量的独立性。

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.