Page 169 - Elements of Distribution Theory

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5.6 Models with a Group Structure 155

The identity element of the group is (1, 0) and since

1 b 1 b

, − (a, b) = (a, b) , − = (1, 0),
a a a a
1 b

−1
(a, b) = , − .
a a
2
The topology on G ls may be taken to be the usual topology on R .
Transformation models
Recall that our goal is to use the group of transformations G to relate different distributions in
P. Consider a random variable X with range X and let G denote a group of transformations
on X. The set of probability distributions P is said to be invariant with respect to G if
the following condition holds: if X has probability distribution P ∈ P, then the probability
distribution of gX, which can be denoted by P 1 ,is also an element of P. That is, for every
P ∈ P and every g ∈ G there exists a P 1 ∈ P such that, for all bounded continuous functions
h : X → R,
E [h(gX)] = E [h(X)]
P
P 1
where E denotes the expectation with respect to P and E P 1 denotes expectation with respect
P
to P 1 .In this case we may write P 1 = gPso that we may view g as operating on P as well
as on X.
We have already considered one example of a class of distributions that is invariant with
respect to a group of transformations when considering exchangeable random variables in
Section 2.6.

Example 5.26 (Exchangeable random variables). Let X 1 , X 2 ,..., X n denote exchange-
ablereal-valuedrandomvariablesandletG denotethesetofallpermutationsof(1, 2,..., n).
Hence, if X = (X 1 ,..., X n ) and g = (n, n − 1,..., 2, 1), for example, then

gX = (X n , X n−1 ,..., X 1 ).

It is straightforward to show that G is a group and, by definition, the set of distributions of
X is invariant with respect to G.

Suppose P is a parametric family of distributions, P ={P(·; θ): θ ∈ }.If P is invariant
with respect to G and if X is distributed according to the distribution with parameter θ,
then gX is distributed according to the distribution with parameter gθ,so that g may be
viewed as acting on the parameter space .In statistics, such a model is often called a
transformation model.

Example 5.27 (Exponential distribution). Let X 1 , X 2 ,..., X n denote independent, iden-
tically distributed random variables, each distributed according to an exponential
distribution with parameter θ> 0. Hence, the vector X = (X 1 ,..., X n ) has density

n

n
p(x; θ) = θ exp −θ x j , x j > 0, j = 1,..., n.
j=1