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 $\displaystyle \hat{p}' = \frac{\hat{p}}{\alpha}$.