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Math Help - Maximum Likehood Estimator

  1. #1
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    Maximum Likehood Estimator

    If f_Y(y;theta) = 2y/(1-theta^2), theta<=y<=1, find the maximum likelihood estimator for theta.

    Any help would be great.
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  2. #2
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    Quote Originally Posted by eigenvector11 View Post
    If f_Y(y;theta) = 2y/(1-theta^2), theta<=y<=1, find the maximum likelihood estimator for theta.

    Any help would be great.
    For an example of what to do, read this: http://www.mathhelpforum.com/math-he...estimator.html
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  3. #3
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by eigenvector11 View Post
    If f_Y(y;theta) = 2y/(1-theta^2), theta<=y<=1, find the maximum likelihood estimator for theta.

    Any help would be great.
    In this case you cannot differentiate.
    The likelihood function is
    L={2^n\prod_{i=1}^ny_i^2\over (1-\theta)^2}<br />
I(\theta \le y_{(1)} \le \cdots \le y_{(n)} \le 1)
    And the largest we can make this is by letting the denominator be as small as possible.
    So we want (1-\theta)^2 small or \theta big.
    BUT by the indicator function the largest \theta can be is
    y_{(1)} our smallest order stat.
    Hence \hat\theta_{MLE}=y_{(1)}
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