Page 174 - Elements of Distribution Theory

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

160 Parametric Families of Distributions

that is, if and only if

T ∗

x 1 = x 2 + 1 (x 1 − x 2 ) /n 1 n = x 2 + b 1 n
n
T
∗
where b = [1 (x 1 − x 2 )]/n.It follows from part (iii) of Theorem 5.6 that T is a maximal
n
invariant.
Consider an invariant statistic T . According to part (ii) of Theorem 5.6, in order to show
that T is not a maximal invariant, it is sufficient to find x 1 , x 2 in X that lie on different orbits
such that T (x 1 ) = T (x 2 ).

4
4
Example 5.34 (Cauchy distribution). Consider X = R and for x ∈ R , x = (x 1 , x 2 ,
x 3 , x 4 ), let
x 1 − x 2
T (x) = .
x 3 − x 4
(4)
In Example 5.32, it is shown that T is invariant with respect to G .
ls
Consider two elements of X, x = (1, 0, 1, 0) and ˜ x = (1, 0, 2, 1). Note that since for
a > 0,
(1, 0, 2, 1) − a(1, 0, 1, 0) = (1 − a, 0, 2 − a, 1)

there does not exist an a such that the elements of x − a ˜ x are all equal. Hence, x and ˜ x lie
on different orbits; see Example 5.30. Since T (x) = T (˜ x) = 1, it follows from part (ii) of
Theorem 5.6 that T is not a maximal invariant.

Equivariance
Consider a group G acting on a set X and let T denote a statistic, T : X → Y for some set
Y. Suppose that G also acts on Y. The statistic T is said to be equivariant if for each g ∈ G,

T (gx) = gT (x), x ∈ X.

Note that two different applications of the transformation g are being used in this expression:
gx refers to the action of g on X, while gT (x) refers to the action of g on Y.
Equivariance is an important concept in statistics. For instance, consider a transformation
model for a random variable X, with respect to a group of transformations G; let X denote
the range of X and let denote the parameter space. Let T denote an estimator of θ,a
function T : X → . Hence, if X = x is observed, we estimate the value of θ to be T (x).
The estimator is equivariant if the estimate corresponding to gx, g ∈ G,is gT (x).

Example 5.35 (Estimation of a location parameter). Let X denote a real-valued random
variable with an absolutely continuous distribution with density p 0 satisfying p 0 (x) > 0,
x ∈ R. Consider the location model based on p 0 consisting of the class of distributions of
(1) (1)
gX, g ∈ G where G denotes the group of location transformations on R.
l l
(1)
Suppose X is distributed according to p 0 and let g = θ denote an element of G . Then
l
the distribution of gX is absolutely continuous with density function
p(x; θ) = p 0 (x − θ), −∞ < x < ∞.