Group family explained

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.^[1]

Consideration of a particular family of distributions as a group family can, in statistical theory, lead to the identification of an ancillary statistic.^[2]

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 $$F\_Y(y)\; =\; P(Y\backslash leq\; y)\; =\; P(X+a\; \backslash leq\; y)\; =\; P(X\; \backslash leq\; y-a)\; =\; F\_X(y-a)$$For a fixed distribution, as a varies from -inf ty to inf ty , 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$$F\_Y(y)\; =\; P(Y\backslash leq\; y)\; =\; P(cX\; \backslash leq\; y)\; =\; P(X\; \backslash leq\; y/c)\; =\; F\_X(y/c)$$

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

$$F\_Y(y)\; =\; P(Y\backslash leq\; y)\; =\; P(cX+a\; \backslash leq\; y)\; =\; P(X\; \backslash leq\; (y-a)/c)\; =\; F\_X((y-a)/c)$$

Note that it is important that $a\; \backslash in\; R$ 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

References

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