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Math Help - unbiased estimators

  1. #1
    Senior Member Danneedshelp's Avatar
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    unbiased estimators

    Q: Suppose that Y_{1},Y_{2},Y_{3} denote a random sample from an exponential distribution with density function

    f(y)=\frac{1}{\theta}e^{-\frac{y}{\theta}} for y>0 and 0 otherwise.

    Consider that following estimators of \theta.

    \theta\hat\\=\frac{Y_{1}+2Y_{2}}{3}

    \theta\hat\\=\overline{Y}

    Note: both thetas directly above should have hats on them.

    I completely stuck on this one. Do I figure the distribution of each function of theta hat and then take the expectation?

    Help getting started on either one would be great.

    Thank you
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  2. #2
    MHF Contributor matheagle's Avatar
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    IS there a question here?
    They both are unbiased.
    BY inspection E(Y_i)=\theta
    Since the coefficients sum to one, both are unbiased.
    Next you should obtain the variances and see that the sample mean has a smaller variance.

    V(\bar Y)={\theta^2\over n} where n=3.

    and V(\hat\theta_2)= {1\over 9}\theta^2+ {4\over 9}\theta^2 = {5\over 9}\theta^2
    Last edited by matheagle; February 10th 2010 at 06:51 PM.
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