Consistency

def. Consisteny. An estimator $\hat{\theta }(X_1,\dots ,X_n)$ is consistent for parameter $\theta$ if:

$$\begin{gathered} \forall ϵ>0,\hat{\theta }\stackrel{n\to \infty }{\underset{p}{⟶}}\theta \\ \text{i.e.} \\ \underset{n\to \infty }{\lim }P(|\hat{\theta }-\theta |<ϵ)=1 \\ \underset{n\to \infty }{\lim }P(|\hat{\theta }-\theta |>ϵ)=0 \end{gathered}$$

![[スクリーンショット 2023-02-21 15.40.01.png|center|250]]

def. Asymptotic Unbiasedness. An estimator $\hat{\theta }$ is asymptotically unbiased if:

$$E[{\hat{\theta }}_n]\stackrel{n\to \infty }{⟶}\theta$$

Remark. You can use the Chebyshev’s Inequality for determining Consistency.