where are known functions. If , then the distribution is said to be in canonical form. When the distribution is in canonical form, the function is called a natural parameter. Other parameters present in the distribution that are not of any interest, or that are already calculated in advance, are called nuisance parameters.
Rearrange the above expression and we have
Set , , , and . Then we see that does indeed belong to the exponential family. Furthermore, it is in canonical form. The natural parameter is .
Similarly, the Poisson, binomial, Gamma, and inverse Gaussian distributions all belong to the exponential family and they are all in canonical form.
Lognormal and Weibull distributions also belong to the exponential family but they are not in canonical form.
If the p.d.f of a random variable belongs to an exponential family, and it is expressed in the second of the two above forms, then
provided that functions and are appropriately conditioned.
Given a member from the exponential family of distributions, we have and , where is the score function and the Fisher information. To see this, first observe that the log-likelihood function from a member of the exponential family of distributions is given by
and hence the score function is
From (1), . Next, we obtain the Fisher information . By definition, we have
On the other hand,
For example, for a Poisson distribution
the natural parameter is and . since Poisson is in canonical form. Then
|Date of creation||2013-03-22 14:30:08|
|Last modified on||2013-03-22 14:30:08|
|Last modified by||CWoo (3771)|
|Defines||canonical exponential family|