# Compare efficiency of several linear combinations of random variables

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

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?

• your formula for the variance of a general linear combination is incorrect. – Glen_b Jul 24 '18 at 10:00
• 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? – Suzanne Jul 24 '18 at 10:39
• $Var(X_3)+2 (a_1 a_3 cov(X_1,X_3)+ a_2 a_3 cov(X_2,X_3))>0$ 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. – Glen_b Jul 24 '18 at 15:09
• 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. – Suzanne Jul 26 '18 at 7:13