# BLUE from calculus

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.

• 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. Feb 16 at 15:14

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.