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Math Help - statistics problem

  1. #1
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    statistics problem

    Let X1,X2,....Xn be a random sample from
    f(x; u , v) = {v/(2*Pi*x^3)}^(1/2) *Exp[-v(x-u)^2/(2*u^2*x)] where u and v are parameters and x>0

    Find a complete sufficient statistic for (u,v) and use it to find the Minimum Variance Unbiased Estimator of u.

    please help me on this one
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  2. #2
    MHF Contributor matheagle's Avatar
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    please type in tex
    and I'll look at it
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  3. #3
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    Let X1,X2,....Xn be a random sample from f(x; u,v)=(v/2\pi x^3)^{1/2}e^{-v(x-u)^2/2u^2x},where  x>0.
    Find a complete sufficient statistic for (u,v) and use it to find the Minimum Variance Unbiased Estimator of u.

    please take a look at it. thank you
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  4. #4
    MHF Contributor matheagle's Avatar
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    If I read this correctly, the exponent needed ()...


    L(x)=\bigg({v\over 2\pi}\biggr)^{n/2} \Pi_{i=1}^n x_i^{-3/2}exp\biggl({-v\over 2u^2}\sum_{i=1}^n(x_i-u)^2/x_i\biggr)

    =\bigg({v\over 2\pi}\biggr)^{n/2} \Pi_{i=1}^n x_i^{-3/2}exp\biggl({-v\over 2u^2}\sum_{i=1}^n(x_i-2u+u^2x_i^{-1})\biggr)

    =\bigg({v\over 2\pi}\biggr)^{n/2} \Pi_{i=1}^n x_i^{-3/2}exp\biggl({-v\sum_{i=1}^nx_i\over 2u^2}+{nv\over u}-{v\sum_{i=1}^nx_i^{-1}\over 2}\biggr)

    So our joint suff stats are \sum_{i=1}^nx_i and \sum_{i=1}^nx_i^{-1}
    Last edited by matheagle; May 1st 2009 at 04:19 PM.
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