# Inverse-variance weighting

I'm writing a Bachelor Thesis in engineering an a small part of it has two deal with assigning weights to independent measurements. This is from the thesis:

I am pretty sure that what equations (3.23) and (3.24) are correct, the fit my data perfectly, but I can't for the life of me find a source to confirm them. I've found two Wikipedia articles relating to them, but they don't really give any sources. Does anyone know whether what I wrote is correct, and if so, where I could find a source to cite?

• This is often (incorrectly, in my opinion) referred to simply as Weighted Least Squares, and a search for that term will find you plenty of sources. – shadowtalker Sep 17 '14 at 14:39
• This result is demonstrated in many threads on this site. To expedite my search I limited it to threads where I posted answers and found stats.stackexchange.com/questions/5392, stats.stackexchange.com/questions/43159, and stats.stackexchange.com/questions/12251. The latter is the most directly relevant one. It corrects two misconceptions apparent in the thesis: (1) it explains how this is the minimum variance unbiased estimator and (2) no assumption of Normality is needed. – whuber Sep 17 '14 at 16:03
• Thanks for the answers. They did make this quite clear. My main problem is still that I can't find a proper source to cite in my thesis. Does anyone know of a statistics book or an article which clearly states that (3.24) is the minimum variance unbiased estimator? – Gunnar Sep 20 '14 at 14:43
• No source needed, I'll just derive 3.24 from 3.23 with the method of Lagrange multipliers – Gunnar Sep 20 '14 at 16:13

For the case of two weights, you can simply set $w_2 = 1 - w_1$, plug it into (3.23) and then find the minimum of the quadratic function of $w_1$.

In general, as explained here on p 5, you have to apply the Lagrange multiplier method that will give you a closed-form solution unless you restrict the weights to be non-negative.