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. Considering a family of distributions as a group family can, in statistical theory, lead to identifying ancillary statistics.
Types
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
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
This family is obtained by multiplying a random variable with a constant. Let be a random variable and be a constant. Let . ThenLocation–scale
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. ThenNote that it is important that and in order to satisfy the properties mentioned in the following section.