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.

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

Formally, consider the vector of random variables

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

where

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

The values X i {\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 X 1 {\displaystyle X_{1}} , and consider the distribution of the remaining intervals within the remaining length. The first element of X {\displaystyle X} , viz X 1 {\displaystyle X_{1}} is defined as neutral if X 1 {\displaystyle X_{1}} is statistically independent of the vector

X 1 ∗ = ( X 2 1 − X 1 , X 3 1 − X 1 , … , X k 1 − X 1 ) . {\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 X 2 {\displaystyle X_{2}} is neutral if X 2 / ( 1 − X 1 ) {\displaystyle X_{2}/(1-X_{1})} is independent of the remaining interval: that is, X 2 / ( 1 − X 1 ) {\displaystyle 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 , … , X k 1 − X 1 − X 2 ) . {\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 X 2 {\displaystyle X_{2}} , viewed as the first element of Y = ( X 2 , X 3 , … , X k ) {\displaystyle Y=(X_{2},X_{3},\ldots ,X_{k})} , is neutral.

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

X 1 , … , j ∗ = ( X j + 1 1 − X 1 − ⋯ − X j , … , X k 1 − X 1 − ⋯ − X j ) . {\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 X = ( X 1 , … , X K ) ∼ Dir ⁡ ( α ) {\displaystyle X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha )} is drawn from a Dirichlet distribution, then X {\displaystyle X} is completely neutral. In 1980, James and Mosimann^[2] showed that the Dirichlet distribution is characterised by neutrality.

• Generalized Dirichlet distribution