Page 225 - Elements of Distribution Theory

P1: JZX
052184472Xc07 CUNY148/Severini May 24, 2005 3:59

7.3 Functions of a Random Vector 211

Equivariance may also be used to simplify the determination of the distribution of a
statistic. Suppose that the distribution of a random variable X is an element of
P ={P(·; θ): θ ∈ }
and that P is invariant with respect to some group of transformations. Suppose that we may
identify the group of transformations with so that if X is distributed according to the
distribution with parameter value e, the identity element of the group, then θ X is distributed
according to P(·; θ).
Suppose that the distributions in P are all absolutely continuous and that the density of
P(·; θ)isgiven by p X (·; θ). Then, by Theorem 7.2,

−1 ∂θ −1
p X (x; θ) = p X (θ x; e) .
x
∂x
In computing the Jacobian here it is important to keep in mind that θ −1 x refers to the
transformation θ −1 applied to x, not to the product of θ −1 and x.
Now let Y = g(X) denote an equivariant statistic so that the set of distributions of Y also
forms a transformation model with respect to . Then we may determine the distribution of
Y under parameter value θ using the following approach. First, we find the density function
of Y under the identity element e, p Y (·; e), using Theorem 7.2. Then the density function
of Y under parameter value θ is given by
∂θ −1

−1
p Y (y; θ) = p Y (θ y .
∂y
y; e)
The advantage of this approach is that, in some cases, it is simpler to apply Theorem 7.2 to
p X (·; e) than to apply it to p X (·; θ) for an arbitrary value of θ ∈ .
Example 7.14 (Difference of uniform random variables). Let X 1 , X 2 denote indepen-
dent, identically distributed random variables, each distributed according to the uniform
distribution on the interval (θ 1 ,θ 2 ), where θ 2 >θ 1 . The uniform distribution on (θ 1 ,θ 2 )is
an absolutely continuous distribution with density function
1
p(x; θ) = ,θ 1 < x <θ 2 .
θ 2 − θ 1
Suppose we are interested in the distribution of X 2 − X 1 .
The family of uniform distributions on (θ 1 ,θ 2 ) with −∞ <θ 1 <θ 2 < ∞ is invariant
under the group of location-scale transformations. Let Z 1 , Z 2 denote independent random
variables each uniformly distributed on the interval (0, 1). Then the distribution of (X 1 , X 2 )
is the same as the distribution of
(θ 2 − θ 1 )(Z 1 , Z 2 ) + θ 1 .

It follows that the distribution of X 2 − X 1 is the same as the distribution of
(θ 2 − θ 1 )(Z 2 − Z 1 );
hence, the distribution of X 2 − X 1 can be obtained by first obtaining the distribution of
Z 2 − Z 1 and then using Theorem 7.1 to find the distribution of X 2 − X 1 .
Let Y 1 = Z 2 − Z 1 and Y 2 = Z 1 . Then the joint density of Y 1 , Y 2 is
1, 0 < y 1 < 1, 0 < y 1 + y 2 < 1.