I understand that the Cramér–Rao bound relates to achieving the lowest possible mean squared error amongst unbiased estimators. Is the same standard used to judge biased estimators? Why/why not?
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1The title of your question suggests you're wanting a proof of why the Cramer-Rao lower bound can't be reached by an estimator of a Gaussian variance (it can in fact, but only when the mean is known); the body suggests you're asking why "efficiency" should be defined as reaching the Cramer-Rao lower bound & what other useful properties of an estimator might be defined. Could you please edit to clarify? (If it's both I think two separate questions would be better.) – Scortchi - Reinstate Monica Jun 03 '15 at 11:22
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Your comment resolved most things for me. I've changed the question to match the second issue you discussed. – user1205901 - Слава Україні Jun 03 '15 at 12:03
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1See Why is the squared difference so commonly used?. I think that leaves a remaining question about why you'd want to restrict your attention to unbiased estimators. – Scortchi - Reinstate Monica Jun 03 '15 at 12:11