Consistent Estimator

Topics: Statistics - Estimator

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(definition)

Let ${\hat{\theta }}_n$ be an estimator for the parameter $\theta$, based on a sample of size $n$.

We say that ${\hat{\theta }}_n$ is consistent if, for any $\epsilon >0$:

$$\underset{n\to \infty }{\lim }\mathbb{P}\left(|{\hat{\theta }}_n-\theta |>\epsilon \right)=0$$

In other words, an estimator based on a sample of size $n$ is consistent when the sequence ${\hat{\theta }}_1,{\hat{\theta }}_2,\dots ,{\hat{\theta }}_n$ converges in probability distribution to $\theta$ as $n\to \infty$.

(proposition)

If ${\hat{\theta }}_n$ is an estimator for $\theta$ and:

$$\underset{n\to \infty }{\lim }\mathbb{E}[{\hat{\theta }}_n]=\theta$$

$$\underset{n\to \infty }{\lim }\mathrm{Var}({\hat{\theta }}_n)=0$$

…then ${\hat{\theta }}_n$ is consistent.