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 [math]\displaystyle{ X }[/math] be a random variable and [math]\displaystyle{ a \in R }[/math] be a constant. Let [math]\displaystyle{ Y = X + a }[/math] . Then [math]\displaystyle{ F_Y(y) = P(Y\leq y) = P(X+a \leq y) = P(X \leq y-a) = F_X(y-a) }[/math]For a fixed distribution , as [math]\displaystyle{ a }[/math] varies from [math]\displaystyle{ -\infty }[/math] to [math]\displaystyle{ \infty }[/math] , the distributions that we obtain constitute the location family.

### Scale Family

This family is obtained by multiplying a random variable with a constant. Let [math]\displaystyle{ X }[/math] be a random variable and [math]\displaystyle{ c \in R^+ }[/math] be a constant. Let [math]\displaystyle{ Y = cX }[/math] . Then[math]\displaystyle{ F_Y(y) = P(Y\leq y) = P(cX \leq y) = P(X \leq y/c) = F_X(y/c) }[/math]

### Location - Scale Family

This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let [math]\displaystyle{ X }[/math] be a random variable , [math]\displaystyle{ a \in R }[/math] and [math]\displaystyle{ c \in R^+ }[/math]be constants. Let [math]\displaystyle{ Y = cX + a }[/math]. Then

[math]\displaystyle{ F_Y(y) = P(Y\leq y) = P(cX+a \leq y) = P(X \leq (y-a)/c) = F_X((y-a)/c) }[/math]

Note that it is important that [math]\displaystyle{ a \in R }[/math] and [math]\displaystyle{ c \in R^+ }[/math] 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)

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