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-   -   Unbiased Estimator of the Standard Error of another Estimator (http://mymathforum.com/advanced-statistics/343368-unbiased-estimator-standard-error-another-estimator.html)

 John Travolski February 8th, 2018 06:48 PM

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?

 romsek February 9th, 2018 01:54 PM

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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