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Math Help - Finding a sufficient statistic

  1. #1
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    Finding a sufficient statistic

    Let f(x|\alpha)=\frac{\Gamma(2\alpha)}{\Gamma(\alpha)^  2}[x(1-x)]^{\alpha -1}

    Then the joint pdf= \prod\limits_{i=1}^n f(x|\alpha)
    = (\frac{\Gamma(2\alpha)}{\Gamma(\alpha)^2})^n \prod\limits_{i=1}^n[x_i(1-x_i)]^{\alpha -1}

    by factorization theorem, g(t,\alpha)h(x)
    t=\prod\limits_{i=1}^n[x_i(1-x_i)]
    g(t,\alpha)=(\frac{\Gamma(2\alpha)}{\Gamma(\alpha)  ^2})^n t^{\alpha -1}
    h(x)=1

    Can I conclude that T(X)=\prod\limits_{i=1}^n[X_i(1-X_i)] is a sufficient statistic?
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  2. #2
    MHF Contributor matheagle's Avatar
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    I assume 0<x<1 and that this is a beta distribution with \alpha =\beta
    I don't think I've seen this question in regards to the beta, but this seems true.

    that matches up with 9 from http://www.ds.unifi.it/VL/VL_EN/point/point6.html
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  3. #3
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    Quote Originally Posted by matheagle View Post
    I assume 0<x<1 and that this is a beta distribution with \alpha =\beta
    I don't think I've seen this question in regards to the beta, but this seems true.

    that matches up with 9 from Sufficient, Complete and Ancillary Statistics
    Thanks just to clarify a bit more, so if \alpha in this case is increased, the distribution will go to normal symmetrical shape. Am I right? thoguh I guess the center is not 0.
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