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Math Help - log likelihood in Pareto distribution

  1. #1
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    log likelihood in Pareto distribution

    I'm having some trouble finding the log-likelihood in these questions:

    Question 1
    Pareto distribution is:
    f(yt) = θ / (yt ^ (1+θ))

    What is the log-likelihood, log L (θ)

    Question 2
    Geometric distribution:
    f (yt) = θ (1 − θ)^yt, yt = 0, 1, 2,
    where θ > 0, is an unknown parameter. Let {y1, y2, ..., yT } be iid observations from the geometric distribution.

    What is the log-likelihood?

    I also need to take derivatives of both. Would appreciate some help, I'm stuck!
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  2. #2
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    Hello,

    Let \mathbf{x}=(x_1,\dots,x_n) a n-sample following a Pareto distribution.

    Then the likelihood function, which is the pdf of the n-tuple, will be :

    L(\theta,\mathbf{x})=\prod_{i=1}^n f(\theta,x_i)=\frac{\theta^n}{\left(\prod_{i=1}^n x_i\right)^{1+\theta}}

    So the log is \log\theta^n-\log \left(\prod_{i=1}^n x_i\right)^{1+\theta}=n\log\theta-(1+\theta)\log \left(\prod_{i=1}^n x_i\right)=n\log\theta-(1+\theta)\sum_{i=1}^n \log(x_i)

    now differentiate with respect to \theta...


    Same thing for the other one.
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  3. #3
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    Thanks Moo! I wasn't so far off Q1 after all.

    Having more trouble with Q2 though, can anyone give me a bit more advice? Thanks
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  4. #4
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    Quote Originally Posted by Denny Crane View Post
    Thanks Moo! I wasn't so far off Q1 after all.

    Having more trouble with Q2 though, can anyone give me a bit more advice? Thanks
    What have you tried? Where are you stuck?
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  5. #5
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    Outsourcing your 306-317 Econometrics homework? Nice.
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  6. #6
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    Quote Originally Posted by jrt12 View Post
    Outsourcing your 306-317 Econometrics homework? Nice.
    If you have evidence of cheating, please report it to a Moderator (electronic copy of assignment, etc.)

    MHF does not knowingly help with questions that count towards a students final grade.

    Thread closed for the time being.
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