Consistency

Weak Consistency

A sequence of estimates ${\displaystyle T_{1},T_{2},...}$ is said to be weakly consistent in probability for ${\displaystyle \psi (\theta )}$if

${\displaystyle \lim _{n\to \infty }P(|T_{n}-\psi (\theta )|\geq \epsilon )=0}$

for every ${\displaystyle \epsilon >0}$

and for every ${\displaystyle \theta \in \Omega }$.

${\displaystyle MSE_{\theta }(T_{n})=E[(T_{n}-\psi (\theta ))^{2}]=var_{\theta }(T_{n})+[E_{\theta }(T_{n})-\psi (\theta )]^{2}}$

${\displaystyle E_{\theta }(T_{n})\rightarrow \psi (\theta )}$

${\displaystyle var_{\theta }(T_{n})\rightarrow 0\Rightarrow {T_{n}}{\xrightarrow {0}}\psi (\theta )}$

${\displaystyle MSE_{\theta }(T_{n})\rightarrow 0}$

Strong Consistency

A sequence of estimates ${\displaystyle T_{1},T_{2},...}$ is said to be strongly consistent with probability 1 for ${\displaystyle \psi (\theta )}$if

${\displaystyle P(\lim _{n\to \infty }T_{n}=\psi (\theta ))=1}$

for every ${\displaystyle \theta \in \Omega }$.

${\displaystyle T_{n}{\xrightarrow {a.s.}}\psi (\theta )\Rightarrow {T_{n}}{\xrightarrow {0}}\psi (\theta )}$