I have a weighted sum, 


weighted sum = w1\*mu1 + (1-w1)\*mu2


with 


variance weighted sum = (w1^2)\*var1 + ((1-w1)^2)\*var2 + 2\*w1\*(1-w1)\*cov


in which


mu1 = mean 1;
mu2 = mean 2;
var1 = variance for mean 1;
var2 = variance for mean 2;
cov = covariance;
w1 = weight (ranging from 0 to 1)


Even though I can compute easily the minimum variance, I am now interested in finding the w1 that gives the largest ratio (weighted sum)/sqrt(variance\_weighted sum).


Do you have any ideas on how one could do that? Any references?


Thanks in advance!


Tiago


\*/


Many thanks, Robert! Does your solution take into account that the denominator is the square root of var\_weighted\_sum? and that the ratio may be either positive or negative? Actually, I am looking for the largest absolute ratio.


The formulation seems to provide weights that do not give the largest ratio. Example:


mu1 = 0.8125358
mu2 = 0.1312268


var1 = 0.123922
var2 = 0.010128


cov = 0.0021274


I know that for this example, the weight (w1) that gives the highest ratio R


where


R = (w1\*mu1 + (1-w1)\*mu2)/sqrt((w1^2)\*var1 + ((1-w1)^2)\*var2 + 2\*w1\*(1-w1)\*cov) 


is w1 = 0.354, giving a R = 2.5867


According to your solution, the w1 = 0.0795, giving a R = 1.885


Since my background is in Biosciences, I will be really grateful if you could comment on that. Am I doing something wrong?



Maximizing the ratio of (weigthed sum)/sqrt(variance_weighted_sum)
It's easy enough to take the derivative and solve for that = 0. The result I get is $w\_1 = \frac{\mu\_2 cov - \mu\_1 var\_2}{(\mu\_1+\mu\_2) cov - \mu\_1 var\_1 - \mu\_2 var\_2}$. This critical point is not necessarily in the interval $[0,1]$ and might be a minimum or inflection rather than a maximum, so the maximum might be here or at 0 or 1. It's also possible that there is no maximum because the variance weighted sum is 0 at some point.


