Group family explained

In probability theory, especially as it is used in statistics, a group family of probability distributions is one obtained by subjecting a random variable with a fixed distribution to a suitable transformation, such as a location–scale family, or otherwise one of probability distributions acted upon by a group.^[1] Considering a family of distributions as a group family can, in statistical theory, lead to identifying ancillary statistics.^[2]

Types

A group family can be generated by subjecting a random variable with a fixed distribution to some suitable transformations. Different types of group families are as follows :

Location

This family is obtained by adding a constant to a random variable. Let

be a random variable and

a\inR

be a constant. Let

Y = X + a

. Then

F_Y(y) = P(Y\leq y) = P(X+a \leq y) = P(X \leq y-a) = F_X(y-a)

For a fixed distribution, as

varies from

-infty

infty

, the distributions that we obtain constitute the location family.

Scale

This family is obtained by multiplying a random variable with a constant. Let

be a random variable and

c\inR⁺

be a constant. Let

Y = cX

. Then

F_Y(y) = P(Y\leq y) = P(cX \leq y) = P(X \leq y/c) = F_X(y/c)

Location–scale

This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let

be a random variable,

a\inR

and

c\inR⁺

be constants. Let

Y=cX+a

. Then

$F_Y(y) = P(Y\leq y) = P(cX+a \leq y) = P(X \leq (y-a)/c) = F_X((y-a)/c)$

Note that it is important that $a \in R$ and

c\inR⁺

in order to satisfy the properties mentioned in the following section.

Transformation

The transformation applied to the random variable must satisfy the properties of closure under composition and inversion.

References

Book: Lehmann, E. L.. George Casella. Theory of Point Estimation. Springer. 1998. 2nd. 0-387-98502-6.
Cox, D.R. (2006) Principles of Statistical Inference, CUP. (Section 4.4.2)