I've heard the term "root-n" consistent estimator' used many times. From the resources I've been instructed by, I thought that a "root-n" consistent estimator meant that:

  • the estimator converges on the true value (hence the word "consistent")
  • the estimator converges at a rate of $1/\sqrt{n}$

This puzzles me, since $1/\sqrt{n}$ doesn't converge? Am I missing something crucial here?

  • 9
    $\begingroup$ It means $\sqrt{n}(\hat\theta-\theta)=O_p(1)$. $\endgroup$
    – hejseb
    Commented Apr 14, 2016 at 6:19
  • $\begingroup$ but $\hat\theta$ is a variable, so how would you compute this? $\endgroup$
    – makansij
    Commented Apr 14, 2016 at 6:44
  • $\begingroup$ @hejseb, I appreciate your response, thank you. Would you please explain in words? It helps me to be able to verbalize, rather than just looking at symbols. $\endgroup$
    – makansij
    Commented Apr 14, 2016 at 6:58
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    $\begingroup$ Nice question! But I'm confused by the claim that $1/\sqrt n$ doesn't converge, what did you mean by that? $\endgroup$
    – Silverfish
    Commented Apr 14, 2016 at 14:27
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    $\begingroup$ You confuse the sequence $1/\sqrt{n} = 1/1, 1/\sqrt{2}, 1/\sqrt{3}, \ldots$ with the series $\sum_{i=1}^n 1/\sqrt{k}$ whose general term is $1/1 + 1/\sqrt{2} + 1/\sqrt{3} + \cdots + 1/\sqrt{n}$. The former converges to $0$ as $n$ grows large whereas the latter diverges. The latter, however, is irrelevant. $\endgroup$
    – whuber
    Commented Apr 14, 2016 at 15:56

1 Answer 1


What hejseb means is that $\sqrt{n}(\hat\theta-\theta)$ is "bounded in probability", loosely speaking that the probability that $\sqrt{n}(\hat\theta-\theta)$ takes on "extreme" values is "small".

Now, $\sqrt{n}$ evidently diverges to infinity. If the product of $\sqrt{n}$ and $(\hat\theta-\theta)$ is bounded, that must mean that $(\hat\theta-\theta)$ goes to zero in probability, formally $\hat\theta-\theta=o_p(1)$, and in particular at rate $1/\sqrt{n}$ if the product is to be bounded. Formally, $$ \hat\theta-\theta=O_p(n^{-1/2}) $$ $\hat\theta-\theta=o_p(1)$ is just another way of saying we have consistency - the error "vanishes" as $n\to\infty$. Note that $\hat\theta-\theta=O_p(1)$ would not be enough (see the comments) for consistency, as that would only mean that the error $\hat\theta-\theta$ is bounded, but not that it goes to zero.

  • $\begingroup$ So, for a an estimator to be "consistent", it must have an $O(1)$, constant value, because if it were $O(n)$, then the estimate would diverge as n increases. $\endgroup$
    – makansij
    Commented Apr 15, 2016 at 4:18
  • 2
    $\begingroup$ No, not quite, see my edit. $\endgroup$ Commented Apr 15, 2016 at 7:33
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    $\begingroup$ I think it's fascinating that an estimator can be consistent and biased, or not consistent and unbiased! $\endgroup$
    – RobertF
    Commented May 17, 2023 at 12:14

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