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# The Exponential Family of Distributions

[Read HC 7.5 where we will use for their and for their .]

Definition 7..4
The exponential family of distributions is a one-parameter family that can be written in the form

 (7.3)

(a)
neither nor depends on ,
(b)
is a non-trivial continuous function of ,
(c)
each of and is a continuous function of , ,
we say that we have a regular case of the exponential family.
Most of the well-known distributions can be put into this form, for example, binomial, Poisson, geometric, gamma and normal. The joint density function of a random sample X from such a distribution can be written as
 (7.4)

Putting

we see that can be written as

so that Theorem 1.1 applies and t(X) is a sufficient statistic for .

Example 7..6
Let X U[0,]. Then , x [0,]. We see that f(x) cannot be written in the form (1.1). We could write B()=1/, p(), but then we would need

which makes h(x) depend on and the condition of (1.1) would not be satisfied.
[We already know that is sufficient for here, and note that is not of the form .]

Example 7..7
Consider the normal distribution with mean and variance . The density function can be written in the form (1.1) where

and . So is minimal sufficient for .

Note that we could have defined and , so that is also sufficient for .

A distribution from the exponential family arises from tilting a simple density ,

and is termed the natural parameter.

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Bob Murison 2000-10-31