Page 167 - Elements of Distribution Theory

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5.6 Models with a Group Structure 153

directly from the corresponding property of the distribution of X.For example, if we know
that
2
4
2
4
4
E(X ) = 3σ + 6µ σ + µ ,
it follows immediately that
2 2
4
4
4
4
2
E(Y ) = 3c σ + 6(cµ + b) c σ + (cµ + b) .
Now consider n independent random variables X 1 , X 2 ,..., X n such that the marginal
distribution of each X j is the same as the distribution of X above. Then we may write
X j = µ + σ Z j , j = 1,..., n, where Z 1 ,..., Z n are independent standard normal random
¯ n ¯ ¯ ¯ n
variables. Let X = X j /n; since X = µ + σ Z where Z = Z j /n,it follows that
j=1 j=1
¯
¯
the relationship between X and Z is the same as the relationship between X j and Z j .For
¯
instance, if we know that Z has a normal distribution with mean 0 and standard deviation
√
¯
1/ n,it follows immediately that X has a normal distribution with mean µ and standard
√
¯
deviation σ/ n. The statistic X is an example of an equivariant statistic.
Consider the statistic
¯
X 1 − X
T = .
¯
X 2 − X
¯
¯
It follows immediately from the facts that X j = µ + σ Z j and X = µ + σ Z that T has the
same distribution as
Z 1 − Z ¯
Z 2 − Z ¯
so that the distribution of T does not depend on µ or σ. Hence, when studying the distri-
butional properties of T we can assume that µ = 0 and σ = 1. This is an example of an
invariant statistic.
The goal of this section is to generalize the ideas presented in the previous example. Let
X denote a random variable with range X and probability distribution P taking values in
a set P. The key idea in Example 5.24 is that there is an algebraic operation on the space
X that corresponds to changes in the distribution of X;we will refer to these algebraic
operations as transformations. Hence, we need to specify a set of transformations on X and
relate them to the different distributions in P.In order to do this, it is convenient to use the
language of group theory.
Consider a transformation g : X → X.We require that g is one-to-one, so that g(x 1 ) =
g(x 2 ) implies that x 1 = x 2 , and onto, so that every x 1 ∈ X may be written g(x 2 ) for some
x 2 ∈ X.We will be interested in a set of such transformations together with an opera-
tion that allows two transformations to be combined. There are a number of conditions
such a combination must satisfy in order for the set of transformations to be useful. For
instance, if two transformations are combined, they must form another transformation and
every transformation must have an inverse transformation such that, if the transformation
is combined with its inverse, an identity tranformation taking each x ∈ X back to x is
formed.
More formally, let G denote a group of transformations g : X → X. Recall that a group
is a nonempty set G together with a binary operation ◦ such that the following conditions