I have two models that represent a physical process. To determine which model is the best, I make some experiments and compare measured data with data predicted by each of the models. The model with the lowest root mean square error ($\sigma$) between measured and predicted data will be selected.
If I make only one experiment, (i.e., I have only one sample) it is trivial to find the best $\sigma$. But I have two experiments, their data cannot be considered to come from the same sample, and both samples should have the same weight in the selection of the model. Then, I have two models: $a$ and $b$, two experiments: $1$ and $2$, and four $\sigma$: $\sigma_{a1}$, $\sigma_{a2}$, $\sigma_{b1}$ and $\sigma_{b2}$.
I have derived that the best model will be the one with the minimum geometric mean of $\sigma$ for both experiments (i.e., the minimum between $\sqrt{\sigma_{a1} . \sigma_{a2}}$ and $\sqrt{\sigma_{b1} . \sigma_{b2}}$).
Am I right? It would be very important for me to find a reference for this result, can you help me?