What are the asymptotic properties of an estimator?

Normally, when we talk about the asymptotic estimator, we talk about the efficiency and consistency of this estimator. I am wondering whether efficiency is AIC and consistency is BIC mentioned in this post? Is there any intuitive way to understand asymptotic properties?

• You may access a larger pool of experts asking the question on Cross Validated Stack Exchange. Sep 25 at 11:52

In simplified and intuitive way:

Consistency is ability of the estimator to on average uncover true value of the coefficient. For example, if the true value of some coefficient $$\beta=2$$ then estimator $$E[\hat{\beta}]=\beta=2$$ as well. An estimator that in expectations would not give you the true beta coefficient would not be consistent.

Efficiency is the ability of estimator to estimate the value of $$\beta$$ as precisely as possible. Due to probabilistic nature of the problem every time you estimate something in statistics there will be some confidence interval. Efficiency, in many standard models that assume normality, means that the estimator has the smaller confidence interval for the true estimate than other possible estimators.

For example, estimator that tells you that $$E[\hat{\beta}] = 3 \pm 0.5$$ is more efficient than estimator that tells $$E[\hat{\beta}] = 3 \pm 1$$.

Moreover, note that estimators can be:

1. Consistent and efficient
2. Consistent but not efficient
3. Inconsistent but efficient
4. Inconsistent and not efficient

All of the above is possible, it all depends on the asymptotic properties of mode you are looking about.

I am wondering whether efficiency is AIC and consistency is BIC mentioned in this post?

No AIC and BIC are information criteria which are estimators in their own right that post there just discusses the properties of AIC and BIC mentioning that the advantage of one is that it is consistent and the other that it is efficient

• Your description of consistency looks more like a description of unbiasedness. Consistency is not about averaging but about convergence. Defining efficiency via confidence intervals seems unorthodox and questionable. Also, what do you mean by $\hat{\beta}= E[\beta]$ where $\beta$ (being the true parameter value) is a constant? And then regarding case 3. inconsistent but efficient, this probably needs some qualification. Would you have an example? (Cc @NguyenLis) Sep 25 at 11:48
• What does $E(\beta)=\beta$ even mean? It is trivially correct but irrelevant. Did you mean $E(\hat\beta)=\beta$ instead? Newbold's quote is neither sufficient nor even necessary (though the latter depends on precise wording) for consistency. I do not think the simplification is justified, as it seems to have introduced some fundamentally wrong ideas about what consistency is. Sep 25 at 19:34
• $\text{Var}(\hat\beta)<\text{Var}(\tilde\beta)$ does not generally imply the confidence interval for $\hat\beta$ is narrowed than for $\tilde\beta$, though under an additional assumption of e.g. normality it would. Sep 25 at 19:37
• I would suggest going back to the definitions and thinking hard how much you can paraphrase them without compromising their meaning. And again, what you present as consistency is unbiasedness, a rather different concept. You may also ask a question on Cross Validated if you have any doubts. The existing questions there already include some fascinating discussions about the subtleties of the notion of consistency. Perhaps also efficiency (not sure about it). Sep 25 at 19:40
• @1mouflon1: unbiasedness and consistency are two independent notions, in the sense that an estimator of $\beta$ can be both biased and consistent for $\beta$, while another can be both unbiased and inconsistent. Sep 26 at 18:34