Theory of Statistics - George Mason University

0.1 Measure, Integration, and Functional Analysis 747
The CDFs of the ɛ-mixture distributions defined on page 185 is a simple
example of a distribution function space. In that space, one of the CDFs is
degenerate.
Important distribution function spaces are those consisting of CDFs P
such that for given p ≥ 1
t p dP < ∞. (0.1.99)
Such a distribution function space is denoted by P p . (Constrast this with the
L p space.) It is clear that P p1 ⊆ P p2 if p1 ≥ p2.
Spaces of distribution functions are related to divisibility of the distributions.
They are useful in robustness studies. Most of the interesting families
of probability distributions as discussed in Chapter 2 do not generate distribution
function spaces.
0.1.11 Transformation Groups
On page 624 we have an example of a group on a set of bijections. Such
transformation groups are important in statistics and are useful in establishing
desirable properties of statistical procedures.
Example 0.1.10 (Continuation of Example 0.0.4) Group of linear
transformations
A common instance of the group G of bijections is formed by functions of the
form
g(x) = bx − c, x, b, c ∈ IR, b = 0.
For given g, we see that g −1 (x) = (x + c)/b ∈ G.
0.1.11.1 Invariant Functions
Definition 0.1.50 (Invariant function)
Let G be a transformation group with domain X. A function f with domain
X is said to be invariant under the transformation group G if for all x ∈ X
and g ∈ G,
f(g(x)) = f(x). (0.1.100)
We also use the phrases “invariant over ...” and “invariant with respect to
...” to denote this kind of invariance.
Example 0.1.11 Invariant function
The function
f(x) = max(d − x 2 )
is invariant over the group G = {g : g(x) = bx − c, x, b, c ∈ IR, b = 0} and
function composition.
Theory of Statistics c○2000–2013 James E. Gentle