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Math Help - Logarithmic distribution: Find the canonical parameter

  1. #1
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    Logarithmic distribution: Find the canonical parameter

    Hi,

    The logarithmic's probability function is:



    To find the canonical parameter, one must re-express the above to the generic pdf of exponential distribution:



    I have managed to:

    1: take log then exp:



    2. apply log rules:



    Unfortunately, this is where I could not expand further to get the expression to be in the generic form. So far, I know (rightly or wrongly):



    but = ???

    Thanks.
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  2. #2
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    Re: Logarithmic distribution: Find the canonical parameter

    Quote Originally Posted by meeksoup View Post
    Hi,

    The logarithmic's probability function is:



    To find the canonical parameter, one must re-express the above to the generic pdf of exponential distribution:



    I have managed to:

    1: take log then exp:



    2. apply log rules:



    Unfortunately, this is where I could not expand further to get the expression to be in the generic form. So far, I know (rightly or wrongly):



    but = ???

    Thanks.
    you're pretty close

    $\ln\left(\dfrac{-1}{\ln(1-p)}\dfrac{p^y}{y}\right)=\ln\left(\dfrac{-1}{\ln(1-p)}\right)+y\ln(p)-ln(y)$

    by inspection

    $\phi=1$

    $\theta=\ln(p)$

    $b(\theta)=-\ln\left(\dfrac{-1}{\ln(1-p)}\right)=-\ln\left(\dfrac{-1}{\ln(1-e^\theta)}\right)$

    $c(y,\phi)=-\ln(y)$
    Thanks from meeksoup
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  3. #3
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    Re: Logarithmic distribution: Find the canonical parameter

    Thanks.
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