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.Cox, D.R. (2006) ''Principles of Statistical Inference'', CUP. (Section 4.4.2)

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 $F_Y(y) = P(Y\leq y) = P(X+a \leq y) = P(X \leq y-a) = F_X(y-a)$For a fixed distribution, as $a$ varies from $-\infty$ to $\infty$, 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$F_Y(y) = P(Y\leq y) = P(cX \leq y) = P(X \leq y/c) = F_X(y/c)$

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

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

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

Parametric statistics
Types of probability distributions

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