Page 175 - Elements of Distribution Theory

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052184472Xc05 CUNY148/Severini May 24, 2005 17:53

5.6 Models with a Group Structure 161

The model given by

{p(·; θ): −∞ <θ < ∞}
(1) (1)
is a transformation model with respect to G .For g ∈ G and θ ∈ = R, gθ = θ + g.
l l
Now consider independent, identically distributed random variables X 1 ,..., X n , each
distributed according to an absolutely continuous distribution with density function of the
form p(·; θ), as given above. The density for (X 1 ,..., X n )isof the form
n

p 0 (x j − θ), −∞ < x j < ∞, i = j,..., n.
j=1
and the model for (X 1 ,..., X n )isa transformation model with respect to G l and gθ = θ + g.
Now consider T ,an estimator of the location parameter θ. The estimator is equivariant
if, for any b ∈ R,

T (x + b1 n ) = T (x) + b;
that is, T is equivariant if adding a constant b to each observation shifts the estimate of θ
by b.

Theorem 5.7. Consider a space X and let G denote a group acting on X.Ifa statistic T is
equivariant then for each x 1 , x 2 ∈ X,
T (x 1 ) = T (x 2 ) implies that T (gx 1 ) = T (gx 2 ) for all g ∈ G.

Conversely, if
T (x 1 ) = T (x 2 ) implies that T (gx 1 ) = T (gx 2 ) for all g ∈ G,

then the action of G on Y, the range of T (X), may be defined so that T is equivariant.

Proof. Let T be an equivariant statistic and suppose T (x 1 ) = T (x 2 ). Let g ∈ G. Then, by
the definition of equivariance,
T (gx i ) = gT (x i ), i = 1, 2.

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

Now suppose that T (gx 1 ) = T (gx 2 ) for all g ∈ G whenever T (x 1 ) = T (x 2 ). For g ∈ G
and x ∈ X, define gT (x) = T (gx). Since T (x 1 ) = T (x 2 ) implies T (gx 1 ) = T (gx 2 ), gT (x)
is well defined. It remains to verify that (T1) and (T2) are satisfied.
Note
eT (x) = T (ex) = T (x),

verifying (T1) and that
g 1 (g 2 T (x)) = g 1 T (g 2 x) = T (g 1 g 2 x) = T ((g 1 g 2 )x) = (g 1 g 2 )T (x),

verifying (T2). The result follows.