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Math Help - How do I find the moment estimator and maximum likelihood from a denstiy function?

  1. #1
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    How do I find the moment estimator and maximum likelihood from a denstiy function?

    Hi Guys,

    Any help will be great.

    Let
    X1;X2; .......;Xn be a random sample from a distribution which has probability density function:


    f
    (x; ) = 2(^2)/[x^3]

    with
    <= x < infinity;


    Find both the moment estimator and the maximum likelihood estimator of the parameter
    .

    Thanks

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  2. #2
    MHF Contributor matheagle's Avatar
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    The likelihood function is {2^nu^{2n}\over \Pi_{i=1}^nx_i^3}I(X_{(1)}\ge u)

    Where that X_{(1)} is the first order stat.
    YOU do not differentiate this. You want the max wrt u.
    The max occurs when u is as large as possible since u is in the numerator.
    But the largest this can be is X_{(1)}, otherwise you have ZERO for that indicator.
    HENCE the MLE of u is X_{(1)}.

    AS for the MOM, I get 2u as the E(X), so get \bar X=\hat E(X)

    or \bar X=2\hat u or \hat u=\bar X/2.
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  3. #3
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    Thanks, do you mind explaining how you got the likelihood function?

    Also for the MOM how did you arrive at 2u for E(X)?

    Cheers.
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  4. #4
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    Quote Originally Posted by matheagle View Post
    The likelihood function is {2^nu^{2n}\over \Pi_{i=1}^nx_i^3}I(X_{(1)}\ge u)

    Where that X_{(1)} is the first order stat.
    YOU do not differentiate this. You want the max wrt u.
    The max occurs when u is as large as possible since u is in the numerator.
    But the largest this can be is X_{(1)}, otherwise you have ZERO for that indicator.
    HENCE the MLE of u is X_{(1)}.

    AS for the MOM, I get 2u as the E(X), so get \bar X=\hat E(X)

    or \bar X=2\hat u or \hat u=\bar X/2.
    heyz, i was wondering how did u manage to get the second part of the likelihood function. i could somehow work out n i=1 2^n theta^ 2n over xi^3
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  5. #5
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by downdown View Post
    heyz, i was wondering how did u manage to get the second part of the likelihood function. i could somehow work out n i=1 2^n theta^ 2n over xi^3

    You need all the x's to exceed u.
    That's the same as the smallest of the data to exceed u.
    Hence the smallest order stat is an important statistic for this parameter.
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