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.

DefinitionEdit

A single element   of a random vector   is neutral if the relative proportions of all the other elements are independent of  .

Formally, consider the vector of random variables

 

where

 

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

 

Variable   is neutral if   is independent of the remaining interval: that is,   being independent of

 

Thus  , viewed as the first element of  , is neutral.

In general, variable   is neutral if   is independent of

 

Complete neutralityEdit

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

If   is drawn from a Dirichlet distribution, then   is completely neutral. In 1980, James and Mosimann[2] showed that the Dirichlet distribution is characterised by neutrality.

See alsoEdit

ReferencesEdit

  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.