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]

## Contents

* 1 Types of group families
* 1.1 Location Family
* 1.2 Scale Family
* 1.3 Location - Scale Family
* 2 Properties of the transformations
* 3 References

## 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 X be a random variable and a∈R be a constant. Let Y=X+a . Then FY(y)=P(Y≤y)=P(X+a≤y)=P(X≤y−a)=FX(y−a)For a fixed distribution , as a 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 X be a random variable and c∈R+ be a constant. Let Y=cX . ThenFY(y)=P(Y≤y)=P(cX≤y)=P(X≤y/c)=FX(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∈R and c∈R+be constants. Let Y=cX+a. Then

FY(y)=P(Y≤y)=P(cX+a≤y)=P(X≤(y−a)/c)=FX((y−a)/c)

Note that it is important that a∈R and c∈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.[1]

* Closure under composition
* Closure under inversion

## References

1. ↑ 1.0 1.1 1.2 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)

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Group family. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Group_family&oldid=178503"