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Thread: 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 $\displaystyle [0,\theta]$. Find an unbiased estimator for $\displaystyle \theta^2$.

    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 $\displaystyle [0,\theta]$. Find an unbiased estimator for $\displaystyle \theta^2$.

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

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

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

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

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

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

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

    This is the unbiased estimator of $\displaystyle \theta^2$.

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