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]

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 :

This family is obtained by adding a constant to a random variable. Let ${\displaystyle X}$ be a random variable and ${\displaystyle a\in R}$ be a constant. Let $Y=X+a$ . Then $${\displaystyle F_Y(y)=P(Y\le y)=P(X+a\le y)=P(X\le y-a)=F_X(y-a)}$$For a fixed distribution , as ${\displaystyle a}$ varies from ${\displaystyle -\mathrm{\infty }}$ to ${\displaystyle \mathrm{\infty }}$ , the distributions that we obtain constitute the location family.

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

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

$${\displaystyle F_Y(y)=P(Y\le y)=P(cX+a\le y)=P(X\le (y-a)/c)=F_X((y-a)/c)}$$

Note that it is important that $a\in R$ and ${\displaystyle c\in R^{+}}$ in order to satisfy the properties mentioned in the following section.

The transformation applied to the random variable must satisfy the following properties.^[1]

• Closure under composition
• Closure under inversion

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