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Math Help - canonical parameter and canonical link function

  1. #1
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    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?
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  2. #2
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    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.
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  3. #3
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    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!!
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  4. #4
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    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.
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