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.
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.
is statistically independent of the vector Variable
is independent of the remaining interval: that is,
, viewed as the first element of
In general, variable
is independent of A vector for which each element is neutral is completely neutral.
is drawn from a Dirichlet distribution, then
In 1980, James and Mosimann[2] showed that the Dirichlet distribution is characterised by neutrality.