My Math Forum Unbiased Estimator of the Standard Error of another Estimator

 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 $\tilde \theta=2\bar Y-1$. 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: 1,757 Thanks: 900 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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