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Math Help - Sufficient estimators question

  1. #1
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    Sufficient estimators question

    A random sample of size n is drawn from the pdf,

    f_Y(y;\theta) = e^{-(y-\theta)}, \theta\le y

    I can show that Y_{max} is not sufficient by an example, but I am having a hard time showing that Y_{min} is sufficient.
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  2. #2
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    So I got that the f_{y_{min}}(y_{min};\theta) = n[e^{-(y-\theta)}]^n, but I can't seem to factor this out of \prod{e^{-(y-\theta)}}

    Hope this helps clarify my question.
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  3. #3
    MHF Contributor matheagle's Avatar
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    It's sufficient by the Factorization Theorem.

    f_Y(y;\theta) = e^{-(y-\theta)}, \theta\le y

    should be rewritten with indicators.

    BUT you need a sample here

    L(y_1,...,y_n) =\prod_{i=1}^n e^{-(y_i-\theta)}I(y_i>\theta)

    =e^{n\theta}e^{-\sum_{i=1}^ny_i}I(y_{min}>\theta)
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  4. #4
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    I am not familiar with what you mean with I? Thanks for your help.
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  5. #5
    Moo
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    Indicator/characteristic function :

    I(y_i>\theta)=1 if y_i is indeed >\theta. 0 otherwise.
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