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Math Help - Sufficiency and Maximum Likelihood.

  1. #1
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    Sufficiency and Maximum Likelihood.

    Let X_1, ... X_n be a random sample from the density function given by,

    f(x;\theta) = (\frac{1}{\theta})rx^{r-1}e^{-x^{r}/\theta}

    0 otherwise, \theta > 0, x > 0 for known r.

    Find a sufficient statistic and maximum likelihood for \theta. Argue that the maximum likelihood estimator is also sufficient for \theta
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  2. #2
    MHF Contributor matheagle's Avatar
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    By the factorization theorem

    \sum_{i=1}^n X_i^r or any multiple of it is suff for \theta

    we get that from the likelihood function.
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  3. #3
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    Then what's the maximum likelihood for theta? Is it possible to illustrate a few steps to kick start? Thank you!
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