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Math Help - Mixture of Gamma, Poison, and Pareto Distribution

  1. #1
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    Mixture of Gamma, Poison, and Pareto Distribution



    I got this far:

    \frac{\Gamma(\alpha + k)}{\Gamma(h) \Gamma(\alpha) \Gamma(k)} \int^{\infty}_0 \int^{\infty}_0 \frac{e^{-(h^{-1} + \lambda) \theta}\theta^{h+x-1} \lambda^{k+x-1} h^{-2k}}{(1+h^{-1} \lambda)^{\alpha + k}}d \lambda d \theta

    now I'm stuck. Not sure how to integrate with respect to lambda. Any help would be greately appreciated.
    Last edited by statmajor; October 6th 2009 at 05:15 PM.
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  2. #2
    MHF Contributor matheagle's Avatar
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    I'm not sure what you're using for a generalized Pareto.
    There are conflicting definitions.
    Likewise there are two definitions of the Gamma.
    I know what the Poisson is, but not the Poison (joke).
    This Pareto is weird lloking to me, mine usually starts at 1, not 0.
    Last edited by matheagle; October 7th 2009 at 08:42 PM.
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  3. #3
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    <br />
f(X|\lambda\vartheta) = \frac{(\lambda\vartheta)^{x}e^{-\lambda\vartheta}}{x!}<br />

    <br />
g(\theta) = \frac{(h^{-1})^{h}}{\Gamma(h)}\theta^{h - 1}e^{-(h^{-1})\theta} <br />

    h(\lambda) = \frac{\Gamma(\alpha + k)h^{-k} \lambda^{k-1}}{\Gamma( \alpha) \Gamma(k)(1+h^{-1}x)^{\alpha + k}}

    then I did this:

    <br />
f_X(x) = \int^{\infty}_0 \int^{\infty}_0f(X|\lambda\vartheta) g(\theta) h(\lambda) d \lambda d\theta<br />
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