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I'm wondering if there is a difference between an asymptotically unbiased estimator and a consistent estimator.

For asymptotically unbiased estimators, the expected value of the estimator converges to the parameter, while for a consistent estimator, the estimator converges in probability to the parameter.

These sound nearly identical, but I believe that being consistent might actually be stronger than being asymptotically unbiased, since being consistent, i.e. converging in probability, implies that not only will the estimates be roughly centered around the true value, but will also be getting closer to it as n increases.

Can someone confirm or refute this claim?

DavidSilverberg
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    Yes, consistency implies asymptotic unbiasedness. – StubbornAtom Jan 19 '19 at 19:15
  • Possible duplicate of [What's the difference between asymptotic unbiasedness and consistency?](https://stats.stackexchange.com/questions/215369/whats-the-difference-between-asymptotic-unbiasedness-and-consistency) Also see https://stats.stackexchange.com/questions/280684/intuitive-understanding-of-the-difference-between-consistent-and-asymptotically?noredirect=1&lq=1. – StubbornAtom Jan 19 '19 at 20:15
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    They are not the same thing. For example, the first observation from a sample is unbiased for the mean regardless of the sample size, but it is not a consistent estimator. – if_the_correlations_are_zero Jan 19 '19 at 21:26

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