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Let $p'\beta$ be an estimable LPF. Suppose that $l'y$ is the candidate which must satisfy the unbiasedness condition and the minimum-variance condition. Formulate this as an optimization problem with Lagrange multipliers, and show that the optimum $l'y$ is $p'(X'X)^{-}X'y$

This is from Jammalamadaka & Sengupta. I'm quite stuck in this one as how should I form the equation for lagrange multipier, I'm sure I can do the rest afterwards. Any help would be appreciated.

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  • $\begingroup$ Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$ Feb 16 at 15:14

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Any linear function of $y$ can be written as $l'y$ for some $l$ by Riesz representation theorem. So you need to check $\mathbb{E}(l'y)=p'\beta$. $$L(\beta,\lambda) = (y-X\beta)'(y-X\beta) + \lambda'(l'X\beta-p'\beta)$$ Take the derivative and set it to zero to get your result. Estimability arument might come into play somewhere.

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