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] Consider random variables <math>X_1,\ldots,X_k</math> where <math>\sum_{i=1}^k X_i=1</math>; interpret the <math>X_i</math> as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say <math>X_1</math>, and consider the distribution of the remaining interval. One then defines the first element of <math>X</math>, viz <math>X_1</math> as neutral if <math>X_1</math> is statistically independent of the vector <math>X_2/(1-X_1),X_3/(1-X_1),\ldots,X_k/(1-X_1)</math>. Variable <math>X_2</math> is neutral if <math>X_2/(1-X_1)</math> is independent of the remaining interval: that is, <math>X_2/(1-X_1)</math> being independent of <math>X_3/(1-X_1-X_2),X_4/(1-X_1-X_2),\ldots,X_k/(1-X_1-X_2)</math>. Thus <math>X_2</math>, viewed as the first element of <math>Y=X_2,X_3,\ldots,X_k</math>, is neutral. In general, variable <math>X_j</math> is neutral if <math>X_1,\ldots X_{j-1}</math> is independent of <math>X_{j+1}/(1-X_1-\cdots -X_j),\ldots X_k/(1-X_1-\cdots - X_j)</math>. A vector for which each element is neutral is completely neutral. If <math>X = (X_1, \ldots, X_K)\sim\operatorname{Dir}(\alpha)</math> is drawn from a Dirichlet distribution, then <math>X</math> is completely neutral.

See also

Generalized Dirichlet distribution

References