Vector-valued estimator

Summary

• Given:
• A sample \(x_1,x_2,...,x_n\)
• A statistical model where \(θ_1,…,θ_m\) are unknown parameters
• If \({\hat{θ}}_i\) is an estimator for \(θ_i\) for \(i=1,…,m\) then we can collect the estimators and unknown parameters in vectors

\[\hat{θ}=\pmatrix {\hat{θ}_1 \\ \vdots \\ \hat{θ}_m} θ=\pmatrix {θ_1 \\ \vdots \\ θ_m}\]

• We say that \(\hat{θ}\) is an unbiased estimator of \(θ\) if

\[E\left( \hat{θ} \right)=θ\]

• If \({\hat{θ}}_1\) and \({\hat{θ}}_2\) are two unbiased estimators of \(θ\) then we say that \({\hat{θ}}_1\) is more efficient than \({\hat{θ}}_2\) if

\[Var\left( {\hat{θ}}_2 \right)-Var\left( {\hat{θ}}_1 \right)\]

• is positive semi-definite.
• We say that \(\hat{θ}\) is a consistent estimator of \(θ\) if

\[\textrm{plim } \hat{θ}=θ\]

• where

\[\textrm{plim } \hat{θ} = \pmatrix {\textrm{plim } \hat{θ}_1 \\ \vdots \\ \textrm{plim }\hat{θ}_m}\]