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When we say an estimator is consistent, we mean "as the sample size increases, sampling distribution of the estimator converges to the true parameter value."

But when we say "an estimator is asymptotically normally distributed", what does it mean?

Are "central limit theorem" and "asymptotically normally distributed" synonymous?

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But when we say "an estimator is asymptotically normally distributed", what does it mean?

Using similar language to your first sentence, when we say an estimator is asymptotically normally distributed, we mean something like as the sample size increases, the sampling distribution of a suitably standardized version of the estimator converges in distribution to some particular normal distribution.

Are "central limit theorem" and "asymptotically normally distributed" synonymous?

Not in general, I think. Some quantity may be asymptotically normal but not come about as a result of any of the versions of the CLT (at least not in any obvious way - it might perhaps be that all of them can ultimately relate to the CLT, but I suspect it's possible to construct cases that would not).

However, very many estimators can be cast as a kind of average of some random variable and in that case a CLT-type argument may be indeed possible.

In some other cases you can combine the CLT with some other result to produce an argument that some estimator should be asymptotically normal (so the CLT may be involved but doesn't stand alone as the basis for the asymptotic normality).

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  • $\begingroup$ I understand that Some quantity may be asymptotically normal but not come about as a result of any of the versions of the CLT (at least not in any obvious way). But if it is CLT, does not it asymptotically normally distributed? $\endgroup$ – user 31466 Dec 7 '16 at 3:45
  • $\begingroup$ I'm sorry but I don't understand what your second sentence is asking. $\endgroup$ – Glen_b -Reinstate Monica Dec 7 '16 at 13:01
  • $\begingroup$ I haven't understood CLT may be involved by doesn't stand alone as the basis for the asymptotic normality. But CLT states that the sampling distribution of the sampling means approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution. Then why CLT alone cannot assure an estimator is asymptotically normal? $\endgroup$ – user 31466 Dec 8 '16 at 0:17
  • $\begingroup$ 1, Sorry, there's a typo there in what you quote. It should say "but" where I typed "by". That probably doesn't help. I will fix it. 2. The CLT gives a way to get asymptotic normality. But that doesn't imply there are no other ways for something to be asymptotically normal. By way of an example, consider $X_i \sim \text{Beta}(\alpha\cdot i,\alpha\cdot i)$. Let $Z_i=(X_i-\mu_i)/\sigma_i$ where $\mu_i$ and $\sigma_i$ are the mean and standard deviation of $X_I$. Then it turns out that $Z_i$ is asymptotically normal but we're not averaging anything... $\endgroup$ – Glen_b -Reinstate Monica Dec 8 '16 at 1:54
  • $\begingroup$ ... In this case (as with many others) it may turn out that with sufficient cleverness one may be able to find a connection to something where one could invoke a CLT type argument --- but that doesn't imply that there is always such an argument. [It would be interesting to know if that could always be done.] $\endgroup$ – Glen_b -Reinstate Monica Dec 8 '16 at 2:03

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