Neutral vector

A single element ${\displaystyle X_{i}}$ of a random vector ${\displaystyle X_{1},X_{2},\ldots ,X_{k}}$ is neutral if the relative proportions of all the other elements are independent of ${\displaystyle X_{i}}$.

Formally, consider the vector of random variables

${\displaystyle X=(X_{1},\ldots ,X_{k})}$

where

${\displaystyle \sum _{i=1}^{k}X_{i}=1.}$

The values ${\displaystyle X_{i}}$ are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say ${\displaystyle X_{1}}$, and consider the distribution of the remaining intervals within the remaining length. The first element of ${\displaystyle X}$, viz ${\displaystyle X_{1}}$ is defined as neutral if ${\displaystyle X_{1}}$ is statistically independent of the vector

${\displaystyle X_{1}^{*}=\left({\frac {X_{2}}{1-X_{1}}},{\frac {X_{3}}{1-X_{1}}},\ldots ,{\frac {X_{k}}{1-X_{1}}}\right).}$

Variable ${\displaystyle X_{2}}$ is neutral if ${\displaystyle X_{2}/(1-X_{1})}$ is independent of the remaining interval: that is, ${\displaystyle X_{2}/(1-X_{1})}$ being independent of

${\displaystyle X_{1,2}^{*}=\left({\frac {X_{3}}{1-X_{1}-X_{2}}},{\frac {X_{4}}{1-X_{1}-X_{2}}},\ldots ,{\frac {X_{k}}{1-X_{1}-X_{2}}}\right).}$

Thus ${\displaystyle X_{2}}$, viewed as the first element of ${\displaystyle Y=(X_{2},X_{3},\ldots ,X_{k})}$, is neutral.

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

${\displaystyle X_{1,\ldots ,j}^{*}=\left({\frac {X_{j+1}}{1-X_{1}-\cdots -X_{j}}},\ldots ,{\frac {X_{k}}{1-X_{1}-\cdots -X_{j}}}\right).}$

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

If ${\displaystyle X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha )}$ is drawn from a Dirichlet distribution, then ${\displaystyle X}$ is completely neutral. In 1980, James and Mosimann^[2] showed that the Dirichlet distribution is characterised by neutrality.

• Generalized Dirichlet distribution