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Neutral vector

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Neutral vector

In statistics, and specifically in the study of the Dirichlet distribution, a neutral vector of random variables is one that exhibits a particular type of statistical independence amongst its elements.[1] In particular, when elements of the random vector must add up to certain sum, then an element in the vector is neutral with respect to the others if the distribution of the vector created by expressing the remaining elements as proportions of their total is independent of the element that was omitted.

Definition

A single element X_i of a random vector X_1,X_2,\ldots,X_k is neutral if the relative proportions of all the other elements are independent of X_i. The concept was originally developed for the study of turtle scutes.

Formally, consider the vector of random variables

X=X_1,\ldots,X_k

where

\sum_{i=1}^k X_i=1..

The values X_i are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say X_1, and consider the distribution of the remaining intervals within the remaining length. The first element of X, viz X_1 is defined as neutral if X_1 is statistically independent of the vector

X^*_1=X_2/(1-X_1),X_3/(1-X_1),\ldots,X_k/(1-X_1).

Variable X_2 is neutral if X_2/(1-X_1) is independent of the remaining interval: that is, X_2/(1-X_1) being independent of

X^*_{1,2}=X_3/(1-X_1-X_2),X_4/(1-X_1-X_2),\ldots,X_k/(1-X_1-X_2).

Thus X_2, viewed as the first element of Y=X_2,X_3,\ldots,X_k, is neutral.

In general, variable X_j is neutral if X_1,\ldots X_{j-1} is independent of

X^*_{1,...,j}=X_{j+1}/(1-X_1-\cdots -X_j),\ldots X_k/(1-X_1-\cdots - X_j).

A vector for which each element is neutral is completely neutral.

If X = (X_1, \ldots, X_K)\sim\operatorname{Dir}(\alpha) is drawn from a Dirichlet distribution, then X is completely neutral. In 1980, James and Mosimann[2] showed that the Dirichlet distribution is characterised by neutrality.


See also

References

  1. ^ Connor, R. J.; Mosimann, J. E. (1969). "Concepts of Independence for Proportions with a Generalization of the Dirichlet Distribution". Journal of the American Statistical Association 64 (325): 194–206.  
  2. ^ James, Ian R.; Mosimann, James E (1980). "A new characterization of the Dirichlet distribution through neutrality". The Annals of Statistics 8 (1): 183–189. 
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