Consistency

Summary

Given:

• A sample $x_1,x_2,...,x_n$
• A statistical model where $θ$ is an unknown parameter
• An estimator ${\hat{θ}}_n=g(x_1,…,x_n)$

Definition: consistency

• We may view ${\hat{θ}}_1,{\hat{θ}}_2,{\hat{θ}}_3,…$ as a sequence of random variables .
• We say that ${\hat{θ}}_n$ is a consistent estimator of $θ$ if ${\hat{θ}}_n$ converges in probability to $θ$ ,

$$\textrm{plim } {\hat{θ}}_n=θ$$

• or

$$P\left( \left| {\hat{θ}}_n-θ \right|<ε \right)→1 \textrm{ as } n→∞$$

• for all $ε>0$ .

Example

• $x_1,x_2,...,x_n$ is an IID random sample where $E\left( x_i \right)=μ$ and $Var\left( x_i \right)=σ^2$ . Then ${\bar{x}}_n$ is a consistent estimator of $μ$ ( the weak law of large numbers ).