Theory of Statistics - George Mason University

180 2 Distribution Theory and Statistical Models
Given a d-variate random variable X, a d × d positive-definite matrix Σ
and a d-vector µ, it is clear that if F(x) is the CDF associated with the random
variable X, then |Σ −1/2 |F(Σ −1/2 (x −µ)) is the CDF associated with Y . The
class of all distributions characterized by CDFs that can be formed in this
way is of interest.
Definition 2.3 (location-scale families)
Let X be a random variable on IR k , let V ⊆ IR k , and let Mk be the collection
of k × k symmetric positive definite matrices. The family of distributions of
the random variables of the form
Y = Σ 1/2 X + µ, for µ ∈ V, Σ ∈ Mk
(2.23)
is called a location-scale family. The group of linear transformations y = g(x)
in equation (2.23) is also called the location-scale group.
The random variable space associated with a location-scale family is a
linear space.
If the PDF of a distribution in a location-scale family is f(x), the PDF
of any other distribution in that family is |Σ −1/2 |f(Σ −1/2 (x − µ)). In the
case of a scalar x, this simplifies to f((x − µ)/σ)/σ. Thus, in a location-scale
family the kernel of the PDF is invariant under linear transformations (see
Definition 0.1.100 on page 747). The probability measure itself is invariant to
the location transformation and equivariant to the scale transformation.
We often use
f((x − µ)/σ)/σ (2.24)
generically to represent the PDF of a distribution in a location-scale family.
While we can always form a location-scale family beginning with any distribution,
our interest is in which of the usual families of distributions are
location-scale families. Clearly, a location-scale family must have enough parameters
and parameters of the right form in order for the location-scale
transformation to result in a distribution in the same family. For example,
a three-parameter gamma distribution is a location-scale family, but a twoparameter
gamma (without the range dependency) is not.
In Table 2.4, I list some common distribution families in which we can
identify a location parameter. While the usual form of the family has more
than one parameter, if all but one of the parameters are considered to be fixed
(that is, effectively, they are not parameters), the remaining parameter is a
location parameter.
An interesting property of a location family is that the likelihood function
is the same as the PDF. Figure 1.2 on page 21 illustrates the difference in a
likelihood function and a corresponding PDF. In that case, the distribution
family was exponential(0, θ), which of course is not a location family. A similar
pair of plots for exponential(α, θ0), which is a location family, would be
identical to each other (for appropriate choices of α on the one hand and x
on the other, of course).
Theory of Statistics c○2000–2013 James E. Gentle