Tests to find which theory agrees better with observation I have three curves ( 1.> observation: yobs , 2.> theory-1: yth1 , 3.> theory-2: yth2 ). All of these curves are functions of a single variable (say variable x.) From a computational perspective, all these curves can be thought of as arrays with discrete values. To get an idea of error bars around  yobs, I use 100 numerical simulations. All arrays, yobs, yth1, yth2 and all 100 simulations have the same length (same number of values).

What statistical test can tell me which one of these two theories (yth1 or yth2)  is a more accurate representation of the observation ( yobs ) inside a given distance range ([x1, x2])?

Below is a schematic diagram of yobs, yth1 and yth2. The orange shaded region around yobs shows  error bar gotten from 100 simulations.



I have already tried  analysis using the formulae given underneath (here,  denotes covariance matrix obtained from the 100 simulations). But, I get very high values of / degree of freedom for comparisons of yth1 with yobs , and yth2 with yobs .  As such, I am looking for tests other than the  test.





Another test that I have tried is the method of fractional errors (shown in the two formulae given below). While the fractional tests tell me which theory is a more accurate model for observations, it does not use errors from 100 simulations. So, I am also trying to find an alternative to this.
I'm not sure about the best statistical way, but one practical approach would be to compare actuals v expected for each of the two hypothetical curves.

Calculating the sum of squared differences (i.e. sum((y_obs - y_theory)^2)) will give you a numerical estimate of goodness-of-fit for each curve. You could also use other metrics depending on your application (e.g. RMSE, RMSLE, SMAPE...).

Or, for each theoretical curve, you could look at the distribution of differences between the actual observation and the value predicted by the curve. It can be handy to view these empirical differences graphically.

I'm sure there are other statistical ways to evaluate goodness-of-fit.