Page 167 - Probability and Statistical Inference

144 3. Multivariate Random Variables

In the two Examples 3.8.4-3.8.5, the support χ depended
on the single parameter θ. But, even if the support χ does
not depend on θ, in some cases the pmf or the pdf may not
belong to the exponential family defined via (3.8.1).

Example 3.8.6 Suppose that a random variable X has the N(θ, θ ) distri-
2
bution where θ(> 0) is the single parameter. The corresponding pdf f(x; θ)
can be expressed as

which does not have the same form as in (3.8.1). In other words, this distri-
bution does not belong to a one-parameter exponential family. !

A distribution such as N(θ, θ ) with θ(> 0) is said to belong to a
2
curved exponential family, introduced by Efron (1975, 1978).

3.8.2 Multi-parameter Situation
Let X be a random variable with the pmf or pdf given by f(x; θθ θθ θ), x ∈ χ⊆ ℜ, θθ θθ θ
= (θ , ..., θ ) ∈ Θ ⊆ ℜ . Here, θθ θθ θ is a vector valued parameter having k compo-
k
1
k
nents involved in the expression of f(x; θθ θθ θ) which is again referred to as a
statistical model.
Definition 3.8.2 We say that f(x; θθ θθ θ) belongs to the k-parameter exponen-
tial family if and only if one can express

with some appropriate forms for g(x) ≥ 0, a(θθ θθ θ) ≥ 0, b (θθ θθ θ) and R (x), i = 1, ...,
i
i
k. It is crucial to note that the expressions of a(θθ θθ θ) and b (θθ θθ θ), i = 1, ..., k, can
i
not involve x, while the expressions of g(x) and R (x), ..., R (x) can not in-
1
k
volve θθ θθ θ.
Many standard distributions, including several listed in Section 1.7, belong
to this rich class. In order to involve only statistically meaningful
reparameterizations while representing f(x; θθ θθ θ) in the form given by (3.8.4),
one would assume that the following regulatory conditions are satisfied: