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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?

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

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  • $\begingroup$ Okay, thanks, I wasn't quite sure if that's what I was going for or not, but now I know. $\endgroup$ Feb 9, 2018 at 18:31

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