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Method of Moments (Beta Dist)

  1. Nov 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]X_1,...,X_n [/itex] be iid with pdf [itex]f(x;\theta) = \theta x^{\theta-1} , 0 \le x \le 1 , 0 < \theta < \infty [/itex]

    Find an estimator for [itex]\theta[/itex] by method of moments


    2. Relevant equations



    3. The attempt at a solution
    I know I need to align the first moment of the beta distribution with the first moment of the sample ([itex]\bar{x}[/itex] or [itex]\dfrac{\Sigma_{i=1}^n x_i}{n}[/itex])

    The beta distribution has a first moment of [itex]\dfrac{\alpha}{\alpha + \beta}[/itex]


    I guess my problem is figuring out what my should be alpha and beta from the given pdf, from there it is simply just [itex]\bar{x}=\dfrac{\alpha}{\alpha + \beta}[/itex] and then solving for [itex]\theta[/itex] Any advice?

    I did try to take the expected value of the pdf and set it equal to x bar, but I think that is not the correct answer ( I got something like [itex]\hat{\theta}=\dfrac{\bar{x}}{1-\bar{x}}[/itex] )

    I also attempted this using MLE and got something like [itex]\hat{\theta}=\dfrac{-n}{\Sigma_{i=1}^n \ln{x_i}} [/itex] but I would also like to solve this problem with method of moments.

    Thanks in advance.
     
    Last edited: Nov 18, 2012
  2. jcsd
  3. Nov 18, 2012 #2

    haruspex

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    If you represent it as a case of a Beta distribution, you get α=θ, β=1, yes? And that gives you the same result as you obtained directly. What makes you think it is wrong?
     
  4. Nov 19, 2012 #3
    Thanks for the reply.

    It just felt wrong to me because it was so far from the MLE estimate, but I know that can happen.
     
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