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I understand that consistency of an estimator is large sample property, but does it make sense to talk about consistency in small samples as well? Can I say about the estimator that it is consistent even though I have a very small sample at hand? And conversely, if I have a consistent estimator (as given), can I say that the sample based on which it was estimated is large?

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Consistency does not depend on sample size; it judges if an estimator formula converges its target value in probability. This means we take the limit as $n$ goes to $\infty$ and get rid of the sample size. If you want to somehow quantify the risk you take at your specific sample size, you can look at the variance of your estimator.

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Consistency is an asymptotic property of an estimator, so it only makes sense in the context of an estimator defined over the sequence of all possible sample sizes. That is, if for any given sample size $n \in \mathbb{N}$ we have some estimator $\hat{\theta}_n: \mathbf{x}_n \rightarrow \Theta$, then this gives us the sequence of estimators:

$$\hat{\theta} \equiv \{ \hat{\theta_n} | n \in \mathbb{N}\}.$$

The property of consistency (strong or weak) is a property that applies to this sequence, and asserts that $\hat{\theta}_n \rightarrow \theta$ as $n \rightarrow \infty$ (in some probabilistic sense that differs for strong and weak consistency). So it does not make sense to talk about consistency in small samples (or in large samples for that matter). Any talk about consistency concerns the properties of an infinite sequence of estimators.

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