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Is there a concept of the magnitude of the bias (I guess that would be the square of the difference)?


Typically bias is a signed quantity; there is plenty more you can say about the bias of an estimator.


Minor correction? I think we want $$P(|I-\hat{I}_n)|>\epsilon)\rightarrow 0$$ I think $\hat{I}_n =1/n$ is a consistent estimator of $I=0$ that doesn't meet this definition.


@ak-47: Yes, this is a very good comment. The actual definition uses $\epsilon$; I left it out because I was trying to (over)simplify. But it really should be in there.