Theory of Statistics - George Mason University

1.1 Some Important Probability Facts 13
our analysis usually results in some general description of the distribution,
rather than in a specification of the distribution.)
We assume that every parametric family is identifiable; that is, P =
{Pθ, θ ∈ Θ} is an identifiable parametric family if it is a parametric family
and for θ1, θ2 ∈ Θ if θ1 = θ2 then Pθ1 = Pθ2.
A family that cannot be indexed in this way might be called a nonparametric
family. The term “nonparametric” is most commonly used to refer to
a statistical procedure, rather than to a family, however. In general terms, a
nonparametric procedure is one that does not depend on strict assumptions
about a parametric family.
Example 1.3 a parametric family
An example of a parametric family of distributions for the measurable space
(Ω = {0, 1}, F = 2 Ω ) is that formed from the class of the probability measures
Pπ({1}) = π and Pπ({0}) = 1 − π. This is a parametric family, namely, the
Bernoulli distributions. The index of the family, π, is called the parameter of
the distribution. The measures are dominated by the counting measure.
Example 1.4 a nonparametric family
An example of a nonparametric family over a measurable space (IR, B) is
Pc = {P : P ≪ ν}, where ν is the Lebesgue measure. This family contains
all of the parametric families of Tables A.2 through A.6 of Appendix A as
well as many other families.
There are a number of useful parametric distributions to which we give
names. For example, the normal or Gaussian distribution, the binomial distribution,
the chi-squared, and so on. Each of these distributions is actually a
family of distributions. A specific member of the family is specified by specifying
the value of each parameter associated with the family of distributions.
For a few distributions, we introduce special symbols to denote the distribution.
We use N(µ, σ 2 ) to denote a univariate normal distribution with
parameters µ and σ 2 (the mean and variance). To indicate that a random
variable has a normal distribution, we use notation of the form
X ∼ N(µ, σ 2 ),
which here means that the random variable X has a normal distribution with
parameters µ and σ 2 . We use
Nd(µ, Σ)
to denote a d-variate normal distribution with parameters µ and Σ
We use
U(θ1, θ2)
to denote a uniform distribution with support [θ1, θ2]. The most common
uniform distribution that we will use is U(0, 1).
Theory of Statistics c○2000–2013 James E. Gentle