In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, and specifically in the study of the Dirichlet distribution, a neutral vector of random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s is one that exhibits a particular type of statistical independence

Statistical independence

In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

 amongst its elements. 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.

Definition

A single element of a random vector is neutral if the relative proportions of all the other elements are independent of . The concept was originally developed for the study of turtle

Turtle

Turtles are reptiles of the order Testudines , characterised by a special bony or cartilaginous shell developed from their ribs that acts as a shield...

 scute

Scute

A scute or scutum is a bony external plate or scale, as on the shell of a turtle, the skin of crocodilians, the feet of some birds or the anterior portion of the mesonotum in insects.-Properties:...

s.

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

Statistical independence

In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

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

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

If is drawn from a Dirichlet distribution, then is completely neutral.