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Thread: statistical test problem

  1. #1
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    statistical test problem

    Hey, I have an interesting problem to solve. Any ideas how to estimate the parameter?
    statistical test problem-todari3.jpg
    Thank you for your help!
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  2. #2
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    Re: statistical test problem

    we have to assume the samples are independently taken.

    let $p_\theta(j)$ be the distribution shown in the problem

    Doing that we have the a posteriori density as

    $p_{ap}(\theta)=p_\theta(0)^2 p_\theta(1)^3 p_\theta(2) = \Large -\frac{\theta ^5}{4}-\frac{\theta ^4}{16}+\frac{5 \theta ^3}{64}+\frac{\theta ^2}{256}-\frac{\theta }{128}+\frac{1}{1024}$

    we can apply the usual process to find extrema, differentiate with respect to $\theta$ and solve for that being $0$

    $\dfrac{dp_{ap}}{d\theta} = \Large -\frac{5 \theta ^4}{4}-\frac{\theta ^3}{4}+\frac{15 \theta ^2}{64}+\frac{\theta }{128}-\frac{1}{128}$

    with roots of $\theta = \{-\dfrac 1 2, -\dfrac 1 5, \dfrac 1 4\}$

    The second derivative test must be applied to see which, if any, are maxima.

    I leave it to you to show that $\theta = -\dfrac 1 5$ is the MAP estimate of $\theta$
    Thanks from Vinod
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  3. #3
    Senior Member Vinod's Avatar
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    Re: statistical test problem

    Quote Originally Posted by romsek View Post
    we have to assume the samples are independently taken.

    let $p_\theta(j)$ be the distribution shown in the problem

    Doing that we have the a posteriori density as

    $p_{ap}(\theta)=p_\theta(0)^2 p_\theta(1)^3 p_\theta(2) = \Large -\frac{\theta ^5}{4}-\frac{\theta ^4}{16}+\frac{5 \theta ^3}{64}+\frac{\theta ^2}{256}-\frac{\theta }{128}+\frac{1}{1024}$

    we can apply the usual process to find extrema, differentiate with respect to $\theta$ and solve for that being $0$

    $\dfrac{dp_{ap}}{d\theta} = \Large -\frac{5 \theta ^4}{4}-\frac{\theta ^3}{4}+\frac{15 \theta ^2}{64}+\frac{\theta }{128}-\frac{1}{128}$

    with roots of $\theta = \{-\dfrac 1 2, -\dfrac 1 5, \dfrac 1 4\}$

    The second derivative test must be applied to see which, if any, are maxima.

    I leave it to you to show that $\theta = -\dfrac 1 5$ is the MAP estimate of $\theta$
    Hello,
    What's the meaning of MAP estimate?
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  4. #4
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    Re: statistical test problem

    Quote Originally Posted by Vinod View Post
    Hello,
    What's the meaning of MAP estimate?
    https://en.wikipedia.org/wiki/Maximu...ori_estimation
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  5. #5
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    Re: statistical test problem

    Quote Originally Posted by Vinod View Post
    Hello,
    What's the meaning of MAP estimate?
    Maximum A Posteori

    The best estimate you can make given the A Posteori distribution
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