# Find an Unbiased Estimator of a Function of a Parameter [duplicate]

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Suppose that $Y_1,...,Y_n$ is an IID sample from a uniform distribution $U(\theta,1)$ The method of moments estimator is $\hat \theta=2\bar Y-1$.

Find an unbiased estimator, call it $\hat \beta$, of the quantity:

$$\frac{1-\theta}{\sqrt{3n}}$$

I noticed that if I take the expected value of this arbitrarily chosen estimator: $$\frac{1-\hat \theta}{\sqrt{3n}}$$ I end up with $$E\left( \frac{1-\hat \theta}{\sqrt{3n}} \right)=E\left( \frac{1-(2\bar Y-1)}{\sqrt{3n}} \right) =\frac{1-\theta}{\sqrt{3n}}$$

So I don't quite understand if the goal here is for the expected value of my estimator to equal EXACTLY $\theta$ or if it's supposed to equal the original function of $\theta$. So this abitrarily chosen estimator of mine; is an an unbiased estimate of the given quantity; is it the $\hat \beta$ I seek?

## marked as duplicate by NCh, Misha Lavrov, Claude Leibovici, Mostafa Ayaz, MarkFeb 19 '18 at 8:56

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## 1 Answer

Let $$\hat{\beta}=\frac{1-\hat \theta}{\sqrt{3n}}.$$ Then as you have shown $$E\hat{\beta}=\frac{1-\theta}{\sqrt{3n}}.$$ Hence $\hat{\beta}$ is an unbiased estimator of $\frac{1-\theta}{\sqrt{3n}}$.

• Okay, thanks, I wasn't quite sure if that's what I was going for or not, but now I know. – whipdeedoo Feb 9 '18 at 18:31