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Math Help - Finding an unbiased estimator

  1. #1
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    Finding an unbiased estimator

    A sample of size 1 is drawn from the unifrom pdf defined over the interval [0,\theta]. Find an unbiased estimator for \theta^2.

    Any help would be great. Thanks.
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  2. #2
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    Quote Originally Posted by jass10816 View Post
    A sample of size 1 is drawn from the unifrom pdf defined over the interval [0,\theta]. Find an unbiased estimator for \theta^2.

    Any help would be great. Thanks.
    If your sample is s, the unbiased estimate of \hat{\theta} = 2s. So a guess for the unbiased estimate of \theta^2 would be \hat{\theta}^2 = 4s^2.

    Recall that E(g(x)) = \int_{-\infty}^\infty g(x) f(x)~dx where f(x) is the probability density function, in this case f(x) = \frac{1}{\theta}. Therefore,

    E(\hat{\theta}^2) = E(4s^2) = \int_0^\theta \frac{4s^2}{\theta}~ds = \left.\frac{4s^3}{3\theta}\right|_{s=0}^{s = \theta} = \frac{4}{3}\theta^2

    But if we want the estimator to be unbiased, we should have

    E(\hat{\theta}^2) - \theta^2 = 0

    Therefore, we see that our initial guess should be modified to be

    \hat{\theta}^2 = 3s^2

    This is the unbiased estimator of \theta^2.

    In general this is a trick you can use to find an unbiased estimator. Make a good guess that will make \alpha E(\hat{p}) - p = 0, then take \hat{p}' = \frac{\hat{p}}{\alpha}.
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