Page 170 - Elements of Distribution Theory

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

156 Parametric Families of Distributions

n
Fora function h : R → R,

n
∞ ∞
n
E[h(X); θ] = ··· h(x)θ exp −θ x j dx 1 ··· dx n .
0 0 j=1
Consider the group of scale transformations G s .For a ∈ G s , using a change-of-variable
for the integral,

∞ ∞
n
n
E[h(aX); θ] = ··· h(ax)θ exp −θ x j dx 1 ··· dx n
0 0 j=1

∞ ∞ θ
n n
= ··· h(x) n exp −(θ/a) x j dx 1 ··· dx n .
0 0 a j=1
It follows that if X has an exponential distribution with parameter θ, then aX has an
exponential distribution with parameter value θ/a and, hence, this model is a transformation
model with respect to G s . The elements of G s are nonnegative constants; fora ∈ G s , the action
of a on is given by aθ = θ/a.

In many cases, the parameter space of a transformation model is isomorphic to the group
of transformations G. That is, there is a one-to-one mapping from G to and, hence, the
group of transformations may be identified with the parameter space of the model. In this
case, the group G may be taken to be the parameter space .
To see how such an isomorphism can be constructed, suppose the distribution of X is
an element of P which is invariant with respect to a group of transformations G. Fix some
element θ 0 of and suppose X is distributed according to the distribution with parameter
θ 0 . Then, for g ∈ G, gX is distributed according to the distribution with parameter value
θ 1 = gθ 0 , for some θ 1 ∈ . Hence, we can write θ 1 for g so that, if X is distributed according
to the distribution with parameter θ 0 , θ 1 X is distributed according to the distribution with
parameter θ 1 . If, for each θ ∈ , there is a unique g ∈ G such that θ = gθ 0 and g 1 θ 0 = g 2 θ 0
implies that g 1 = g 2 , then and G are isomorphic and we can proceed as if = G. The
parameter value θ 0 may be identified with the identity element of G.

Example 5.28 (Exponential distribution). Consider the exponential distribution model
considered in Example 5.27; for simplicity, take n = 1. Let g = a > 0 denote a scale
transformation. If X has a standard exponential distribution, then gX has an exponential
distribution with parameter θ 1 = 1/a.
The group G may be identified with using the correspondence a → 1/θ.If X has a
standard exponential distribution, then θ X has an exponential distribution with parameter θ.
Hence, θ X = X/θ; the identity element of the group is the parameter value corresponding
to the standard exponential distribution, 1. The same approach may be used for a vector
(X 1 ,..., X n ).

Example 5.29 (Location-scale models). Let X denote a real-valued random variable with
an absolutely continuous distribution with density p 0 satisfying p 0 (x) > 0, x ∈ R. The
(1)
location-scale model based on p 0 consists of the class of distributions of gX, g ∈ G ls
(1)
where G denotes the group of location-scale transformations on R.
ls