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I want to estimate a scaling factor $s$ between two vectors of scalars $a$ and $b$. In the ideal case it would be constant for all components, so $a = s \cdot b$, however there is some noise in the data.

So I calculate individual samples of the components, like $s_1 = a_1 / b_1$, $s_2 = a_2 / b_2$, etc. and want to estimate the mean and variance of the samples to get an estimate of the best fitting scaling factor and its variance for the whole vectors. Because of the distribution, I use the geometric mean, but how would one correctly estimate the variance in this case? The a and b are very much correlated and always positive. I don't know how they are distributed.

Thanks for any hints.

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