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Sorry in advance; I could not get the markdown editor to output what I wanted. So all I have is this LaTex syntax. We have Rank(X)=p (full rank), and we know that $\hat{\beta}=([X^TX]^{-1}X^T Y)$ We also know that $\mathbb{E}(\hat{\beta})= \beta$ is an unbiased estimator. We are asked to prove that $\lambda^ T \beta$ is estimable for any $\lambda \in \mathbb{R}^p$. I'm kind of stuck, but here are some other results I've either proven earlier in the HW or given to us as a fact: $\lambda^ T \beta$ is estimable iff $\lambda ^T \in R(X)$ $R(X)=R(X^TX)=C(X^TX))$ $\lambda^T \in \ R(X^TX)$ iff $\lambda^TGX^TX=\lambda^ T$, where $G$ is any generalized inverse of $X^TX$ I'm kind of stuck here. Any ideas on what direction I can take this in? Should I use the first fact I listed to prove the $\iff$ statement?
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Fri, 24 Jan 2025 19:04 GMT