Consider the following: $$$$ Suppose that $Y_1,Y_2,...,Y_n$ is an IID sample from a Uniform$[\theta,1]$ distribution. It is easy to show by method of moments that $\hat \theta = 2\bar Y-1$. Find and unbiased estimator of $\sigma_{\hat\theta}$ and show that the estimator is unbiased. $$$$ I would like to know how to approach this problem as the topic of bias/unbiased estimators is really confusing me, if you could explain estimators with both a formal and informal definition/description that would be a plus. Also I showed that the standard error of $\hat\theta$ was $\sigma_{\hat\theta}=\frac{1-\theta}{\sqrt{3n}}$ by taking the variance of $\hat \theta$... but I'm not sure why I took the variance of $\hat \theta$.

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    $\begingroup$ You have shown that the parameter of interest $\sigma_{\hat{\theta}}$ is linear in $\theta$, therefore for any unbiased estimator of $\theta$, $\tilde{\theta}$, $\frac {1 - \tilde{\theta}} {\sqrt{3n}}$ will be an unbiased estimator of $\sigma_{\hat{\theta}}$. Have you found it by checking the expectation? $\endgroup$ – BGM Feb 8 '18 at 6:10
  • $\begingroup$ It's clear that the expectation of $2/bar Y -1$ is $/theta$, is that all that is needed to show unbiasedness? $\endgroup$ – Griffy Feb 8 '18 at 6:37
  • $\begingroup$ Yes, of course. $\endgroup$ – BGM Feb 8 '18 at 10:43

First of all, I hope everything is correct I'm saying. To show unbiasedness E($2\bar Y -1$) should equal $\theta$.

E($2\bar Y -1$) = $\frac{2}{n}$E(Y) $-1$ = $\frac{2}{n}$(E($Y_{1}$) + ... + E($Y_{n}$)) $-1$. We know the expectation of this uniform distribution is $\frac{\theta + 1}{2}$, therefore $\frac{2}{n}$($\frac{n(\theta + 1)}{2}$) $-1$ = $\frac{2n(\theta + 1)}{2n}$ $-1$ = $\theta$. So it is an unbiased estimator.


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