Consistent Estimator

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 (statistic) is considered as consistent, it becomes more reliable with large sample ($n \to \infty$). 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 $\varepsilon$;
\[limit_{n \rightarrow \infty} P[|\hat{\theta}_n-\theta| \le \varepsilon]=1\]
\[limit_{n\rightarrow \infty} P[|\hat{\theta}_n-\theta|> \varepsilon]=0]\]

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

Consistency of an estimator

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.

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.

A consistent estimate has insignificant (non-significant) errors (variations) as sample sizes increases indifinately. More specifically, the probability that those errors will vary by more than a given amount approaches zero as the sample size increases. In other words, the more data you collect, a consistent estimator will be close to the real population parameter you’re trying to measure. The sample mean ($\overline{X}$) and sample variance ($S^2$) are two well-known consistent estimators.