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I have the following linear combinations of (non-independent) random variables:

$Y1=X1+X2$ and $Y2=X1+X2+X3$

With the following general formula, I can calculate the variance for combinations $Var(Y)=\sum_i^n a_i^2 Var(X_i)+2\mathop{\sum\sum}_\limits{i<j} a_i a_j cov(X_i,X_j)$

Imagine I want to know whether I should use $Y1$ or $Y2$ if I want the most efficient linear combination. I can see that $Var(Y2)>Var(Y1)$ if $a_3^2Var(X_3)+2 (a_1 a_3 cov(X_1,X_3)+ a_2 a_3 cov(X_2,X_3))>0$, but that would give me the absolute variance that does not take the range/size of the linear combination into account (which likely differs if $X_3 \neq 0$). I think I need some form of relative variance.

I know I can calculate the efficiency of two estimators, given the same parameter. It may be a very basic question, but is there a formal way to relate the efficiencies of two different linear combinations?

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  • $\begingroup$ your formula for the variance of a general linear combination is incorrect. $\endgroup$ – Glen_b Jul 24 '18 at 10:00
  • $\begingroup$ You're right. I forgot the weights for the covariance and corrected them. Thank you for notifying me. Did you spot any other mistakes that I overlooked? $\endgroup$ – Suzanne Jul 24 '18 at 10:39
  • $\begingroup$ $Var(X_3)+2 (a_1 a_3 cov(X_1,X_3)+ a_2 a_3 cov(X_2,X_3))>0$ either the first term needs a coefficient of $a_3^2$ or you need to use the fact that the $a$'s are all $1$ throughout. $\endgroup$ – Glen_b Jul 24 '18 at 15:09
  • $\begingroup$ Actually, all $\alpha$'s of 1 is the situation I am interested in, so I think that's why I forgot. Corrected it again, to allow some more generalizability. $\endgroup$ – Suzanne Jul 26 '18 at 7:13

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