Page 147 - Elements of Distribution Theory

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5.2 Parameters and Identifiability 133

given by

P(A; F) = dF(x)
A
where F is a distribution function on the real line satisfying F(0) ≥ 1/2 and F(0−) ≥ 1/2.
Let F denote the set of all nondecreasing, right-continuous functions on R satisfying

lim F(x) = 0, lim F(x) = 1, F(0) ≥ 1/2, and F(0−) ≥ 1/2.
x→−∞ x→∞
Then the elements of P may be written P(·; F) where F ∈ F so that F is the parameter
space for the model.

In parametric statistical inference, the set of possible distributions is assumed to be
parameterized by a finite-dimensional parameter so that is a subset of finite-dimensional
Euclidean space; such a model is said to be a parametric model. Models that are not
parametric are said to be nonparametric. The model described in Example 5.1 is a parametric
2
model, with a subset of R ; the model described in Example 5.2 is nonparametric. Here
we will focus on parametric models.
In a parametric model for a random variable Y with parameter θ, all quantities based on
the probability distribution of Y will depend on the value of θ under consideration. When
we wish to emphasize this we will include the parameter in the notation of these quantities;
for instance, we will write probabilities as Pr(·; θ) and expectations as E(·; θ).

Example 5.3 (Normal distributions). Consider a random variable Y with a normal distri-
bution with mean µ and standard deviation σ, where −∞ <µ< ∞ and σ> 0, and let
θ = (µ, σ). Then
2
2
E(Y; θ) = µ, E(Y ; θ) = µ + σ 2
and the characteristic function of Y is given by

2
σ 2
ϕ(t; θ) = exp − t + µit , −∞ < t < ∞.
2
Although in the discussion above models have been described in terms of probabil-
ity distributions, we may equivalently describe the model in terms of distribution func-
tions, density or frequency functions when these exist, or even characteristic functions.
In many cases, either all the distributions in P are absolutely continuous or all are dis-
crete with the same minimal support. In these cases, we describe the model in terms of
either the density functions or the frequency functions; such a function is called the model
function of the model. Furthermore, we will generally describe such a model informally,
by specifying the model function and the parameter space, without explicit construction
of the set P.

Example 5.4 (Normal distributions). Consider the set of normal distributions considered
in Examples 5.1 and 5.3. A more informal way of describing this set is the following: