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Math Help - bias/unbiased estimator

  1. #1
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    bias/unbiased estimator

    having trouble with question in bold, any help will be appreciated.. Thanks!
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  2. #2
    Guy
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    Just look for a function of \hat{\theta} that is unbiased for 1 / \theta. A good start would be 1 / \hat{\theta} and then fix it up so that it's unbiased. Running through this really quick, I got

    \hat{\phi} = \frac{2n - 1}{2n} \frac{1}{\hat{\theta}}

    which you can easily verify is unbiased and consistent. It'll help you out in doing this if you first verify the following: for Y \sim G(\alpha, \beta), under the parametrization of the gamma given,

     \displaystyle<br />
\mbox{E}[Y^r] = \frac{\Gamma(\alpha + r) \beta^r}{\Gamma(\alpha)}<br />

    for all r such that \alpha + r > 0.
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  3. #3
    MHF Contributor matheagle's Avatar
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    for consistency (weak or strong) use the weak or strong law of large numbers for the sample mean.
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