Page 173 - Elements of Distribution Theory

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5.6 Models with a Group Structure 159

Let G denote a group of transformations from X to X and suppose that P is invariant with
k
respect to G. Let T denote a function from X to R .
The following conditions are equivalent:
(i) T is a maximal invariant
(ii) Let x 1 , x 2 denote elements of X such that x 2 /∈ O(x 1 ). Then T is constant on O(x i ),
i = 1, 2 and T (x 1 ) = T (x 2 ).
(iii) Let x 1 , x 2 ∈ X.T (x 1 ) = T (x 2 ) if and only if there exists g ∈ G such that x 1 = gx 2 .

Proof. We first show that conditions (ii) and (iii) are equivalent. Suppose that condition
(ii) holds. If T (x 1 ) = T (x 2 ) for some x 1 , x 2 ∈ X, then we must have x 2 ∈ O(x 1 ) since
otherwise condition (ii) implies that T (x 1 ) = T (x 2 ). Since T is constant on O(x i ), it follows
that T (x 1 ) = T (x 2 )if and only if x 2 ∈ O(x 1 ); that is, condition (iii) holds.
Now suppose condition (iii) holds. Clearly, T is constant on O(x) for any x ∈ X. Fur-
thermore, if x 1 , x 2 are elements of X such that x 2 /∈ O(x 1 ), there does not exist a g such
that x 1 = gx 2 so that T (x 1 ) = T (x 2 ). Hence, condition (ii) holds.
We now show that condition (iii) and condition (i) are equivalent. Suppose that condition
(iii) holds and let T 1 denote an invariant statistic. T is maximal invariant provided that T 1
is a function of T . Define a function h as follows. If y is in the range of T so that y = T (x)
for some x ∈ X, define h(y) = T 1 (x); otherwise, define h(y) arbitrarily. Suppose x 1 , x 2 are
elements of X such that T (x 1 ) = T (x 2 ). Under condition (iii), x 1 = gx 2 for some g ∈ G
so that T 1 (x 1 ) = T 1 (x 2 ); hence, h is well defined. Clearly, h(T (x)) = T 1 (x)so that T is a
maximal invariant. It follows that (iii) implies (i).
Finally, assume that T is a maximal invariant, that is, that (i) holds. Clearly, x 2 = gx 1
implies that T (x 1 ) = T (x 2 ). Suppose that there does not exist a g ∈ G satisfying x 2 = gx 1 .
k
Define a statistic T 1 as follows. Let y 1 , y 2 , y 3 denote distinct elements of R .If x ∈ O(x 1 ),
T 1 (x) = y 1 ,if x ∈ O(x 2 ), T 1 (x) = y 2 ,if x is not an element of either O(x 1 )or O(x 2 ), then
T 1 (x) = y 3 . Note that T 1 is invariant. It follows that there exists a function h such that
T 1 (x) = h(T (x)), x ∈ X,so that h(T (x 1 )) = h(T (x 2 )); hence, T (x 1 ) = T (x 2 ). Therefore
condition (iii) holds.

Theorem 5.6 gives a useful description of a maximal invariant statistic. The range X of a
random variable X can be divided into orbits. Two points x 1 , x 2 lie on the same orbit if there
exists a g ∈ G such that x 2 = gx 1 .Aninvariant statistic is constant on orbits. An invariant
statistic T is a maximal invariant if, in addition, it takes different values on different orbits.
Hence, a maximal invariant statistic completely describes the differences between the orbits
of X;however, it does not give any information regarding the structure within each orbit.

n
Example 5.33 (Location group). Let X = R and consider the group of location transfor-
mations on X.In Example 5.31 it was shown that the function
T

T (x) = x −
1 x /n 1 n
n
is invariant.
n
Suppose that x 1 , x 2 are elements of R . Then T (x 1 ) = T (x 2 )if and only if
T T

x 1 − 1 x 1 /n 1 n = x 2 − 1 x 2 /n 1 n ,
n n