Group family

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.^[1] Different types of group families are as follows :

Location

This family is obtained by adding a constant to a random variable. Let ${\displaystyle X}$ be a random variable and ${\displaystyle a\in R}$ be a constant. Let ${\textstyle Y=X+a}$ . Then $${\displaystyle F_{Y}(y)=P(Y\leq y)=P(X+a\leq y)=P(X\leq y-a)=F_{X}(y-a)}$$For a fixed distribution, as ${\displaystyle a}$ varies from ${\displaystyle -\infty }$ to ${\displaystyle \infty }$, the distributions that we obtain constitute the location family.

Scale

This family is obtained by multiplying a random variable with a constant. Let ${\displaystyle X}$ be a random variable and ${\displaystyle c\in R^{+}}$ be a constant. Let ${\textstyle Y=cX}$ . Then$${\displaystyle F_{Y}(y)=P(Y\leq y)=P(cX\leq y)=P(X\leq y/c)=F_{X}(y/c)}$$

Location–scale

This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let ${\displaystyle X}$ be a random variable, ${\displaystyle a\in R}$ and ${\displaystyle c\in R^{+}}$be constants. Let ${\displaystyle Y=cX+a}$. Then

$${\displaystyle F_{Y}(y)=P(Y\leq y)=P(cX+a\leq y)=P(X\leq (y-a)/c)=F_{X}((y-a)/c)}$$

Note that it is important that ${\textstyle a\in R}$ and ${\displaystyle c\in R^{+}}$ 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.^[1]

References

1. Lehmann, E. L.; George Casella (1998). Theory of Point Estimation (2nd ed.). Springer. ISBN 0-387-98502-6.
2. Cox, D.R. (2006) Principles of Statistical Inference, CUP. ISBN 0-521-68567-2 (Section 4.4.2)