Page 177 - Elements of Distribution Theory

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5.6 Models with a Group Structure 163

so that

θ e = gT 2 (x 1 )θ 1 = g.

Hence, g = e, the identity element of G, and x 1 = x 2 .

Therefore, under the conditions of Theorem 5.8, a random variable X may be written as
(T 1 (X), T 2 (X)) where the distribution of T 1 (X) does not depend on θ and the distribution
of T 2 (X)varies with θ in an equivariant manner.

Example 5.37 (Location model). As in Example 5.35, let (X 1 ,..., X n ) denote a random
vector with density function of the form
n
n
p 0 (x j − θ), x = (x 1 ,..., x n ) ∈ R
j=1
where p 0 is a density function on R. Consider the model corresponding to θ ∈ = R.
This model is invariant under the group of location transformations, as described in
Example 5.25.
In Example 5.33 it is shown that the statistic
n
1
T
T 1 (x) = (x 1 − ¯ x,..., x n − ¯ x) , ¯ x = x j ,
n
j=1
isamaximalinvariant;thestatistic T 2 (x) = ¯ x isequivariant,since T 2 (x + θ1 n ) = T 2 (x) + θ,
n
and the range of T 2 is . Hence, an observation x ∈ R may be described by ¯ x, together
with the residual vector T 1 (x).

Under the conditions of Theorem 5.8, the random variable X is equivalent to a maximal
invariant statistic T 1 and an equivariant statistic T 2 .Itis important to note that these statistics
˜
are not unique, even taking into account the equivalence of statistics. Let T 2 (X) and T 2 (X)
−1 ˜
denote two equivariant statistics and let h(x) = T 2 (x) T 2 (x). Then, for any g ∈ G,
−1 ˜
−1 ˜
˜
h(gx) = T 2 (gx) T 2 (gx) = g −1 T 2 (x) gT 2 (x) = gg −1 T −1 T 2 (x) = h(x).
2
−1 ˜
It follows T 2 (X) T 2 (X)isaninvariant statistic and, hence, a function of T 1 (x). That is,
˜
T 2 (X)is not a function of T 2 (X) alone.
These points are illustrated in the following example.
Example 5.38 (Exponential random variables). As in Example 5.28, let X =
(X 1 ,..., X n ) where X 1 ,..., X n are independent, identically distributed random variables,
each with an exponential distribution with mean θ, θ ∈ = R .As shown in Exam-
+
+ n
ple 5.28, this model is invariant under the group of scale transformations. Here X = (R )
and x, ˜ x ∈ X are in the same orbit if x = a ˜ x for some a > 0.
For x = (x 1 ,..., x n ) ∈ X, let T 1 (x) = (x 2 /x 1 , x 3 /x 1 ,..., x n /x 1 ). Clearly, T 1 is invariant
and it is easy to see that if x = a ˜ x for some a > 0, then T 1 (x) = T 1 (˜ x); hence, T 1 (X)isa
maximal invariant statistic.