Theorem: If "θ hat" is an unbiased estimator for θ AND Var(θ hat)->0 as n->∞, then it is a consistent estimator of θ.(adsbygoogle = window.adsbygoogle || []).push({});

The textbook proved this theorem using Chebyshev's Inequality and Squeeze Theorem and I understand the proof.

BUT then there is a remark that we can replace "unbiased" by "asymptotically unbiased" in the above theorem, and the result will still hold, but the textbook provided no proof. This is where I'm having a lot of trouble. I really don't see how we can prove this (i.e. asymptotically unbiased and variance->0 implies consistent). I tried to modify the original proof, but no way I can get it to work under the assumption of asymptotically unbiased.

I'm frustrated and I hope someone can explain how to prove it. Thank you!

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# Asymptotically unbiased & consistent estimators

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