Page 171 - Elements of Distribution Theory

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

5.6 Models with a Group Structure 157

(1)
Suppose X is distributed according to p 0 and let g = (σ, µ) denote an element of G .
ls
Then

∞ ∞
E[h(gX)] = h(gx)p 0 (x) dx = h(σ x + µ)p 0 (x) dx
−∞ −∞
∞

x − µ 1
= h(x)p 0 dx.
σ σ
−∞
Hence, the distribution of gX is absolutely continuous with density function
1 x − µ
p(x; θ) = p 0 , −∞ < x < ∞.
σ σ
The model given by
{p(·; θ): θ = (σ, µ),σ > 0, −∞ <µ< ∞}
(1)
is a transformation model with respect to G .
ls
For instance, consider the case in which p 0 denotes the density of the standard normal
distribution. Then the set of distributions of X is simply the set of normal distributions with
mean µ and standard deviation σ.
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
1 x i − µ
p 0 , −∞ < x i < ∞, i = 1,..., n.
n
σ σ
Clearly, the model for (X 1 ,..., X n )isa transformation model with respect to G ls .
Let x ∈ X. The orbit of x is that subset of X that consists of all points that are obtainable
from x using a transformation in G; that is, the orbit of x is the set
O(x) ={x 1 ∈ X: x 1 = gx for some g ∈ G}.

n
Example 5.30 (Location-scale group). Let X = R and consider the group of location-
n
scale transformations on X. Then two elements of R , x 1 and x 2 , are on the same orbit if
there exists (a, b) ∈ G ls such that
x 1 = ax 2 + b1 n ,
that is, if there exists a constant a > 0 such that the elements of x 1 − ax 2 are all equal.
Foragiven element x ∈ X,
O(x) ={x 1 ∈ X: x 1 = ax + b1 n , a > 0, −∞ < b < ∞}.

Invariance
k
Now consider a function T : X → R .We say that T (X)isan invariant statistic with respect
to G if, for all g ∈ G and all x ∈ X, T (gx) = T (x). That is, T is constant on the orbits of X.
Hence, if T is invariant, then the probability distribution of T (gX)is the same for all g ∈ G.
In particular, if the family of probability distributions of X is invariant with respect to G then
the distribution of T (X)is the same for all P ∈ P. Thus, in order to find the distribution of
T (X)we may choose a convenient element P ∈ P and determine the distribution of T (X)
under P.