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Math Help - Maximum likelihood estimator

  1. #1
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    Maximum likelihood estimator

    5. A random sample of size n is taken from a distribution with pdf:

    f(x;θ) = θx^(θ-1 ) 0 < x < 1; θ > 0.

    Find the maximum likelihood estimator of θ.


    I tried teaching myself out of the book but it has no examples. Can someone run me through a step by step way of solving this? I have nothing to reference and my notes have 2 lines on it.
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  2. #2
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    Quote Originally Posted by wolverine21 View Post
    5. A random sample of size n is taken from a distribution with pdf:

    f(x;θ) = θx^(θ-1 ) 0 < x < 1; θ > 0.

    Find the maximum likelihood estimator of θ.


    I tried teaching myself out of the book but it has no examples. Can someone run me through a step by step way of solving this? I have nothing to reference and my notes have 2 lines on it.
    The likelihood function L(x_1, x_2, \, .... \, x_n) is defined to be the joint pdf of the random variables X_1, \, X_2, \, .... \, X_n.

    Therefore L(x_1, x_2, \, .... \, x_n) = \left( \theta x_1^{\theta - 1} \right) \cdot \left( \theta x_2^{\theta - 1} \right) \cdot .... \left( \theta x_n^{\theta - 1} \right) = \theta^n \, ( x_1 \cdot x_2 \cdot .... x_n)^{\theta - 1}.

    The maximum likelihood estimate of \theta is the value of \theta that maximises L(x_1, x_2, \, .... \, x_n).

    Since \ln L is a monotonically increasing function of L, both L and \ln L will be a maximum for the same value of \theta. It's obviously easier in this instance to find the value of \theta that maximises \ln L:

    \ln L = n \ln \theta + (\theta - 1) \ln (x_1 \cdot x_2 \cdot .... x_n)

    \Rightarrow \frac{d \ln L}{d\theta} = \frac{n}{\theta} + \ln (x_1 \cdot x_2 \cdot .... x_n).

    Now solve \frac{dL}{d\theta} = 0 for \theta.
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  3. #3
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    Please see my computation.. I don't know if it is correct. I have a question, Is there a possibility that it we can a negative in MLE?...thanks!
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  4. #4
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    Quote Originally Posted by aadbaluyot View Post
    Please see my computation.. I don't know if it is correct. I have a question, Is there a possibility that it we can a negative in MLE?...thanks!
    You are missing a log in the second line. See my post above.
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