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

Scale

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

Location–scale

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, $a\in R$ and $c\in R^{+}$be constants. Let $Y=cX+a$. Then

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

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

Transformation

The transformation applied to the random variable must satisfy the properties of closure under composition and inversion.

References

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