I have a weighted sum,
weighted sum = w1*mu1 + (1-w1)*mu2
variance weighted sum = (w1^2)*var1 + ((1-w1)^2)*var2 + 2*w1*(1-w1)*cov
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!
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
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?