There is always a lower bound for an unbiased estimator called Cramer-Rao Lower Bound. Does any one remember any upper bound for unbiased estimator? The upper bound is used for worst-case analysis of my estimator performance in telecommunication engineering.
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$\begingroup$ pls see the edit $\endgroup$– Seyhmus GüngörenNov 22, 2013 at 10:12
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Such a bound does not make sense. Take $x(t)=A+B\sin(2\pi f_0 t)$ and let your estimator be $\Theta{'}(t)=x(t)$ for some uniformly distributed $t\in[0,T]$. Then, the expected value of your estimator $\Theta{'}(t)$ can be made the same with the true parameter $\Theta$ through increasing (or decreasing) $A$. Hence, this estimator will be unbiased. When $B\rightarrow \infty$, The variance will also go to infinity. As a result one cannot talk about any upperbound of an unbiased estimator.
If additionally the estimator should be minimum variance estimator (MVUnbiasedE), then this estimator is unique therefore such a comparison can also not be done among MVUEs, because there is only one.
answered Nov 21, 2013 at 14:26
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$\begingroup$ One can be more general. An estimator is just a (Borel) measurable function of data. Unbiased-ness fixes a mean $\theta$. It's trivial to construct random variables with a given mean and arbitrarily large variance. $\endgroup$– MichaelNov 22, 2013 at 11:29
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$\begingroup$ @Michael An estimator is not a (Borel) measurable function of data. But as you said one can still be more general. What I gave is just an example t demonstrate this. $\endgroup$ Nov 22, 2013 at 11:41
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$\begingroup$ ...a measurable function that is a statistic whose range lie in the parameter space $\endgroup$– MichaelNov 22, 2013 at 13:12