Group family

In probability theory, especially as that field is used in statistics, a group family of probability distributions is a family obtained by subjecting a random variable with a fixed distribution to a suitable family of transformations such as a location-scale family, or otherwise a family of probability distributions acted upon by a group.

Consideration of a particular family of distributions as a group family can, in statistical theory, lead to the identification of an ancillary statistic.

Types of group families
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 Family
This family is obtained by adding a constant to a random variable. Let $$X$$ be a random variable and $$a \in R$$ 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 $$a $$ varies from $$-\infty $$ to $$\infty $$, the distributions that we obtain constitute the location family.

Scale Family
This family is obtained by multiplying a random variable with a constant. Let $$X$$ be a random variable and $$c \in R^+$$ 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 Family
This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let $$X $$ be a random variable, $$a \in R$$ and $$c \in R^+$$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 $$c \in R^+ $$ in order to satisfy the properties mentioned in the following section.

Properties of the transformations
The transformation applied to the random variable must satisfy the following properties.
 * Closure under composition
 * Closure under inversion