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.

kmcrane

@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.

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.