3.9.2 Estimable Functions

Prove that $\mathbf{a}^{T} \mathbb {E}\left[\widehat{\mathbf{\beta }}\right]$ is an estimable function of $\mathbf{\beta }$.

If $\mathbf{a}_{1}^{T} \beta , \mathbf{a}_{2}^{T} \beta , \ldots , \mathbf{a}_{k}^{T} \beta$ are estimable, prove that any linear combination of these is also estimable.

If $\mathbf{a}^{T} \widehat{\mathbf{\beta }}$ is invariant with respect to $\widehat{\mathbf{\beta }}$, prove that $\mathbf{a}^{T} \mathbf{\beta }$ is estimable.

Prove that $\mathbf{a}^{T} \mathbf{\beta }$ is estimable if and only if

If $\mathbf{a}^{T} \mathbf{\beta }$ is an estimable function, prove that

Prove that all linear functions $\mathbf{a}^{T} \mathbf{\beta }$ are estimable if and only if the columns of $\mathbf{X}$ are linearly independent.