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

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