# canonical parameter and canonical link function

• Apr 6th 2011, 04:01 AM
noob mathematician
canonical parameter and canonical link function
Given the exponential distribution: $f(x)= \theta^{-1}e^{-x/\theta$.

Converts it into exponential family form: $exp[\frac{(1/\theta)x-log(1/\theta)}{-1}]$

Isn't the canonical parameter and link function the same?

Which is $\frac{1}{\theta}$

Are there any differences between canonical parameter and canonical link function?
• Apr 6th 2011, 06:51 AM
theodds
No, this isn't right. You need to move the $-1$ in the denominator upstairs (if you are thinking of this as an exponential dispersion family, you can't have the dispersion parameter be negative). The canonical parameter for this should be $\lambda = \frac {-1} \theta$. The canonical link turns out to be the same - except as a function of the mean instead of the canonical parameter. This isn't true in general, of course (unless your base parameterization is always in terms of the mean, I suppose).

The exponential dispersion form of the density should be: $\exp\left\{(-1 / \theta) x - \log \theta)\right\}$.

It's also worth mentioning that people apparently give the canonical link of the exponential as $1 / \mu$ instead of $-1 / \mu$ because these two are equivalent in terms of actually fitting the model and the negative sign looks ugly. But the canonical parameter DOES have the negative in it.
• Apr 6th 2011, 09:36 AM
noob mathematician
Thanks for all ur reply to all my queries (Include the one on rats too).

I define my exponential family to be in this form:

$exp[\frac{Y\theta-a(\theta)}{\phi}+b(Y,\phi)]$ Will it make a difference to the conclusion?

To further verify it, it turns out that $b'(\theta)$ and $b''(\theta)\phi$ do fit well as $\theta$ and $\theta^2$

So do u mean $\theta$ is the parameter and $a(\theta)$ is the link function??

Sorry I very confused, it will help a lot if u can state the parameter and link function accordingly. Thanks!!
• Apr 6th 2011, 09:41 AM
theodds
You can't have $\phi = -1$. In some sense $\phi$ is a variance parameter so you don't let that be negative. Every source I've ever seen takes $\phi > 0$. Usually this is called an exponential dispersion family, as opposed to an exponential family, but they are closely related.

You should have $\lambda = \frac {-1} \theta$ as the canonical parameter, and $g(\theta) = \frac {-1} \theta$ the canonical link.