A statistics is a consistent estimator of a population parameter if “as the sample size increases, it becomes almost certain that the value of the statistics comes close (closer) to the value of the population parameter”. If an estimator is consistent, it becomes more reliable with large sample. All this means that the distribution of the estimates become more and more concentrated near the value of the population parameter which is being estimated, such that the probability of the estimator being arbitrarily closer to $\theta$ converges to one (sure event).

The estimator $\hat{\theta}_n$ is said to be a consistent estimator of $\theta$ if for any positive $\varepsilon$;

\[limit_{n \rightarrow \infty} P[|\hat{\theta}_n-\theta| \le \varepsilon]=1\]

or

\[limit_{n\rightarrow \infty} P[|\hat{\theta}_n-\theta|> \varepsilon]=0]\]

Here $\hat{\theta}_n$ expresses the estimator of $\theta$, calculated by using a sample size of size $n$.

The sample median is a consistent estimator of the population mean, if the population distribution is symmetrical; otherwise the sample median would approach the population median not the population mean.

The sample estimate of standard deviation is biased but consistent as the distribution of $\hat{\sigma}^2$ is becoming more and more concentrated at $\sigma^2$ as the sample size increases.

A sample statistic can be an inconsistent estimator, whereas a consistent statistic is unbiased in the limit but an unbiased estimator may or may not be consistent estimator.

Note that these two are not equivalent: (1) Unbiasedness is a statement about the expected value of the sampling distribution of the estimator, while (ii) Consistency is a statement about “where the sampling distribution of the estimator is going” as the sample size

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