I have two simple weighted means (WM), say:
WM1 = m1*w1+m2*w2
WM2 = m1*w3+m2*w4
m1 = sample mean 1
m2 = sample mean 2
w1, w2 = weights, so that w1+w2 = 1
w3, w4 = weights, so that w3+w4 = 1
both m1 and m2 have estimated sample variances var1 and var2, respectively.
In addition, both WM1 and WM2 have estimated variances wvar1 and wvar2, respectively.
Since I have only marginal distributions, Is there a way to compute a covariance estimate between WM1 and WM2, that is, cov(WM1,WM2)?
Thanks for any suggestions.