# Thread: Finding an unbiased estimator

1. ## 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.

2. Originally Posted by jass10816
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 $\hat{p}' = \frac{\hat{p}}{\alpha}$.