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Math Help - Finding the Complete Sufficient Statistic

  1. #1
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    Finding the Complete Sufficient Statistic

    Let f(x; \theta) = \theta e^{-\theta x}. Find the complete sufficient statistic for theta.

    From what I understand, I have write the PDF in the "form" of the Regulat Exponential Class/Family which has the PDF of f(x; \theta) = e^{p(\theta )K(x) + S(X) + q(x)}

    f(x;\theta) = e^{-\theta x + log \theta}

    so I know p( \theta) = - \theta, K(x) = x S(x) = 0 and q( \theta ) = log \theta

    I don't understand my textbook's explaination of how to find the complete sufficient statistic, would someone please explain the next step I have to take?

    Thank you.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Clearly \sum X_i is suff for theta, next prove it is complete.
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  3. #3
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    The only thing my book says is:

    The statistic Y_1 = \Sigma X_i is a sufficient statistic for theta and f_{Y_1}(y_1; \theta) is complete, then so is Y1.

    I have no idea what that means.
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