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

Location Family

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

For a fixed distribution , as varies from to , the distributions that we obtain constitute the location family.

Scale Family

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

Location - Scale Family

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 , and be constants. Let . Then

Note that it is important that and 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.[1]

  • Closure under composition
  • Closure under inversion

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)
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