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 {\displaystyle X} be a random variable and a ∈ R {\displaystyle a\in R} be a constant. Let Y = X + a {\textstyle Y=X+a} . Then F Y ( y ) = P ( Y ≤ y ) = P ( X + a ≤ y ) = P ( X ≤ y − a ) = F X ( y − a ) {\displaystyle 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 {\displaystyle a} varies from − ∞ {\displaystyle -\infty } to ∞ {\displaystyle \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 {\displaystyle X} be a random variable and c ∈ R + {\displaystyle c\in R^{+}} be a constant. Let Y = c X {\textstyle Y=cX} . Then F Y ( y ) = P ( Y ≤ y ) = P ( c X ≤ y ) = P ( X ≤ y / c ) = F X ( y / c ) {\displaystyle 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 {\displaystyle X} be a random variable, a ∈ R {\displaystyle a\in R} and c ∈ R + {\displaystyle c\in R^{+}} be constants. Let Y = c X + a {\displaystyle Y=cX+a} . Then
F Y ( y ) = P ( Y ≤ y ) = P ( c X + a ≤ y ) = P ( X ≤ ( y − a ) / c ) = F X ( ( y − a ) / c ) {\displaystyle 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 ∈ R {\textstyle a\in R} and c ∈ R + {\displaystyle 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.
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