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February 8th, 2018, 07:48 PM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  Unbiased Estimator of the Standard Error of another Estimator
Suppose that $Y_1,...,Y_n$ is an IID sample from a uniform $U(\theta, 1)$ distribution. The method of moments estimator for $\theta$ is $\tilde \theta=2\bar Y1$. The standard error of $\tilde \theta$ is $$\sigma_{\tilde \theta}=\frac{1\theta}{\sqrt{3n}}$$ Find an unbiased estimator of $\sigma_{\tilde \theta}$ and show that it is unbiased. So normally I know how to find the estimator of a parameter of a distribution, but I don't quite understand how I'm supposed to find an estimator of the standard deviation of another estimator of a parameter of a distribution... so I really don't know how to go about this. What do I do? 
February 9th, 2018, 02:54 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230 
this turned out to be so simple that I'm just embarrassed about the effort it took. $\begin{align*} &\sigma_{\tilde{\theta}} \approx \dfrac{1  (2\overline{Y}1)}{\sqrt{3n}} =\\ \\ &\dfrac{2(1\overline{Y})}{\sqrt{3n}} ; \\ \\ &E[\sigma_{\tilde{\theta}}] = \dfrac{2\left(1\frac{1+\theta}{2}\right)}{\sqrt{3n}} = \\ \\ &\dfrac{1\theta}{\sqrt{3n}} \end{align*}$ 

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error, estimator, standard, unbiased 
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