Page 168 - Elements of Distribution Theory

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154 Parametric Families of Distributions

are satisfied

(G1)If g 1 , g 2 ∈ G then g 1 ◦ g 2 ∈ G
(G2)If g 1 , g 2 , g 3 ∈ G then (g 1 ◦ g 2 ) ◦ g 3 = g 1 ◦ (g 2 ◦ g 3 )
(G3) There exists an element e ∈ G, called the identity element, such that for each g ∈ G,
e ◦ g = g ◦ e = g
(G4)For each g ∈ G, there exists an element g −1 ∈ G such that g ◦ g −1 = g −1 ◦ g = e.

We assume further that, for a suitable topology on G, the operations

(g 1 , g 2 ) → g 1 ◦ g 2
and g → g −1 are continuous. Often, the group G can be taken to be finite-dimensional
Euclidean space with the topology on G taken to be the usual one. For simplicity, we will
suppress the symbol ◦ when writing group operations so that, for example, g 1 g 2 = g 1 ◦ g 2 .
So far, we have only described how the transformations in G must relate to each other.
However, it is also important to put some requirements on how the transformations in G act
on X.
Let g 1 , g 2 ∈ G. Since g 2 x is an element of X,we may calculate g 1 (g 2 x). We require that
the result is the same as applying g 1 g 2 to x; that is,
(T1) g 1 (g 2 x) = (g 1 g 2 )x.

We also require that the identity element of the group, e,is also the identity transformation
on X:

(T2) ex = x, x ∈ X.

Example 5.25 (Location and scale groups). In statistics, the most commonly used trans-
n
formations are location and scale transformations. Here we take X = R .
+
First consider the group G s of scale transformations. That is, G s = R such that for any
a ∈ G s and x ∈ X, ax represents scalar multiplication. It is easy to see that this is a group
with the group operation defined to be multiplication; that is, for a 1 , a 2 in G s , a 1 a 2 denotes
the multiplication of a 1 and a 2 . The identity element of the group is 1 and a −1 = 1/a.For
the topology on G s we may take the usual topology on R.
We may also consider the set G l of location transformations. Then G l = R and for b ∈ G l
and x ∈ X, bx = x + b1 n where 1 n denotes the vector of length n consisting of all 1s. The
group operation is simple addition, the identity element is 0, b −1 =−b, and the topology
on G l may be taken to be the R-topology.
+
Now consider the group of location–scale transformations, G ls . The group G ls = R × R
such that for any (a, b) ∈ G ls and any x ∈ X,
(a, b)x = ax + b1 n .
Let (a 1 , b 1 ) and (a 2 , b 2 ) denote elements of G ls . Then

(a 1 , b 1 )[(a 2 , b 2 )x] = a 1 (a 2 x + b 2 1 n ) + b 1 1 n = a 1 a 2 x + (a 1 b 2 + b 1 )1 n .
Hence, the group operation is given by

(a 1 , b 1 )(a 2 , b 2 ) = (a 1 a 2 , a 1 b 2 + b 1 ).