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
[edit]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
[edit]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
[edit]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
[edit]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
[edit]The transformation applied to the random variable must satisfy the following properties.[1]
- Closure under composition
- Closure under inversion
References
[edit]- ^ a b c Lehmann, E. L.; George Casella (1998). Theory of Point Estimation (2nd ed.). Springer. ISBN 0-387-98502-6.
- ^ Cox, D.R. (2006) Principles of Statistical Inference, CUP. ISBN 0-521-68567-2 (Section 4.4.2)