Definition:Consistent Estimator

Definition

Let $X_1, X_2, \ldots, X_n$ be random variables.

Let the joint distribution of $X_1, X_2, \ldots, X_n$ be indexed by a population parameter $\theta$.

Let $\hat \theta$ be an estimator of $\theta$.

Then $\hat \theta$ is consistent if and only if:

$\ds \lim_{n \mathop \to \infty} \map \Pr {\size {\hat \theta - \theta} \ge \epsilon} = 0$

for all $\epsilon > 0$.

Also see

• Results about consistent estimators can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): consistent estimator
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): estimator
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): consistent estimator
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): estimator