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Math Help - unbiased est for gamma

  1. #1
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    unbiased est for gamma

    X1,...Xn ~ Gamma(lambda,theta)
    lambda = shape parameter
    theta = scale parameter

    we assume that a<lambda*n

    I need to prove that:

    T=[(gf(n*lambda))/(gf(n*lambda-a))]*[sum(xi)]^-a

    is an unbiased estimator of the function T(theta) = theta^a

    I have no clue how to start it, any clue will be appreciated. I can type it again as a picture if you fail to understand the question.

    thanks
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  2. #2
    MHF Contributor matheagle's Avatar
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    what is gf?
    what does * mean?
    sum xi?
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  3. #3
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    sorry about that...

    gf means gamma function ( I forgot to mention it )
    * is the regular use, multiplication
    sum(Xi) = a sum ( sigma ) of the random variables Xi

    ( picture attached )
    Attached Thumbnails Attached Thumbnails unbiased est for gamma-q007.jpg  
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  4. #4
    MHF Contributor matheagle's Avatar
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    You need to learn TeX or explain what your newly invented terms mean.

    First of all W=\sum_{i=1}^n X_i \sim\Gamma (n\lambda ,\theta ).

    Next, obtain E(W^{-a}) by manipulating the constants.

    You never need to integrate these type of moments.
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  5. #5
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    thanks for your help, it's appreciated !
    sorry for not explaining the terms, I forgot

    can you be a bit more specific about "manipulating the constants" ? what do you mean by that, what should I do ?
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  6. #6
    MHF Contributor matheagle's Avatar
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    E(W^{-a})=\int_0^{\infty} w^{-a}f(w)dw ....
    continue... and incorporate the w^{-a} with the other power of w in the integrand.
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