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 ${\textstyle Y=X+a}$ . 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 ${\textstyle Y=cX}$ . 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 ${\textstyle a\in R}$ 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