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.

## Contents

* 1 Definition
* 1.1 Complete neutrality
* 2 See also
* 3 References

## Definition

A single element Xi of a random vector X1,X2,…,Xk is neutral if the relative proportions of all the other elements are independent of Xi.

Formally, consider the vector of random variables

X=(X1,…,Xk)

where

∑i=1kXi=1.

The values Xi are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say X1, and consider the distribution of the remaining intervals within the remaining length. The first element of X, viz X1 is defined as neutral if X1 is statistically independent of the vector

X1*=(X21−X1,X31−X1,…,Xk1−X1).

Variable X2 is neutral if X2/(1−X1) is independent of the remaining interval: that is, X2/(1−X1) being independent of

X1,2*=(X31−X1−X2,X41−X1−X2,…,Xk1−X1−X2).

Thus X2, viewed as the first element of Y=(X2,X3,…,Xk), is neutral.

In general, variable Xj is neutral if X1,…Xj−1 is independent of

X1,…,j*=(Xj+11−X1−⋯−Xj,…,Xk1−X1−⋯−Xj).

### Complete neutrality

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

If X=(X1,…,XK)∼Dir⁡(α) 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

* Generalized Dirichlet distribution

## 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. doi:10.2307/2283728.
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. doi:10.1214/aos/1176344900.

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