Page 176 - Elements of Distribution Theory

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162 Parametric Families of Distributions

Based on Theorem 5.7, it is tempting to conclude that T is equivariant if and only if
T (x 1 ) = T (x 2 ) implies that T (gx 1 ) = T (gx 2 ). Note, however, that this statement does not
require any specification of the action of G on T (X). Theorem 5.7 states that there is a
definition of the action of G such that T is equivariant. The following simple example
illustrates this point.

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Example 5.36. Let X = R and let G = R .For g ∈ G define gx to be multiplication of
g and x. Let T (x) =|x|. Clearly, T (x 1 ) = T (x 2 ) implies that T (gx 1 ) = T (gx 2 ). However,
equivariance of T depends on how the action of G on T (X)is defined. Suppose that
gT (x) = T (x)/g. Then T is not equivariant since T (gx) =|gx| while gT (x) =|x|/g.
However, we may define the action of G on T (X)ina way that makes T equivariant. In
particular, we may use the definition gT (x) = T (gx)so that, for y ∈ T (X), gy denotes
multiplication of g and y.

Consider a random variable X with range X and distribution in the set

P ={P(·; θ): θ ∈ }.
Let G denote a group of transformations such that P is invariant with respect to G and that
G and are isomorphic.
Let T 1 denote a maximal invariant statistic. Then, as discussed above, T 1 (x) indicates
the orbit on which x resides. However, T 1 does not completely describe the value of x
because it provides no information regarding the location of x on its orbit. Let T 2 denote
an equivariant statistic with range equal to ; hence, the action of a transformation on the
range of T 2 is the same as the action on G. Then T 2 (x) indicates the position of x on its
orbit. The following theorem shows that T 1 (x) and T 2 (x) together are equivalant to x.

Theorem 5.8. Let X denote a random variable with range X and suppose that the distri-
bution of X is an element of

P ={P(·; θ): θ ∈ }.

Let G denote a group of transformations from X to X and suppose that P is invariant with
respect to G. Suppose that G and are isomorphic.
m
Let T : X → R ,T = (T 1 , T 2 ), denote a statistic such that T 1 is a maximal invariant
and T 2 is an equivariant statistic with range . Then T is a one-to-one function on X.

Proof. The theorem holds provided that T (x 1 ) = T (x 2 )if and only if x 1 = x 2 . Clearly,
x 1 = x 2 implies that T (x 1 ) = T (x 2 ); hence, assume that T (x 1 ) = T (x 2 ).
Since T 1 (x 1 ) = T 1 (x 2 ), x 1 and x 2 lie on the same orbit and, hence, there exists g ∈ G
such that x 2 = gx 1 .By the equivariance of T 2 ,

T 2 (x 1 ) = T 2 (x 2 ) = T 2 (gx 1 ) = gT 2 (x 1 ).

Note that T (x 1 ) may be viewed as an element of G. Let θ 1 = T 2 (x 1 ) −1 and let θ e denote
the identity element of . Then

θ e = T 2 (x 1 )θ 1 = gT 2 (x 1 )θ 1