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Thread: Finding the Complete Sufficient Statistic

  1. #1
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    Finding the Complete Sufficient Statistic

    Let $\displaystyle f(x; \theta) = \theta e^{-\theta x}$. 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 $\displaystyle f(x; \theta) = e^{p(\theta )K(x) + S(X) + q(x)}$

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

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

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

    I have no idea what that means.
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