A Monte-Carlo simulation could proceed as follows.

(1) Select a random subset of your equations. You might try different criteria e.g. a few at a time or perhaps 30% at a time. I do not know enough about this area to have a good suggestion here.

(2) Do a fitting to get the best-fit parametervalues for {a,b}.

(3) Attempt to combine your results or at least check that they tend to give consistent results.

Having said this, it's probably overkill. It is not at all clear why one would not just do a best fit to the full set of data. One reason to use subsets, in general, is to avoid overfitting when there is a plethora of parameters. But you only have two. In such situations I think it is fairly common to either use the full set, or at worst use a large subset and then "test" the result by checking how well it "predicts" the remaining cases.

The upshot is I think there are basic questions of the science and statistics involved that you will need to consider before you can really pose this as a coding question.

Dear Nasser,

Thanks for your comment. The a and b are unknowns and my final goal is to find the best fit of a and b. I don't know what supposed to be the expected best estimation values of a and b, but I can know whether the obtained a and b are good or not by doing validation. For my study, those a and b are runoff parameters to estimate/ to simulate runoff.

I know the real value of runoff from the observation data (observed runoff). So if I have a pair of a and b, I can simulate the runoff and then overplotted the simulated runoff with the observed runoff. The smallest discrepancy (error) between the simulated runoff and observed runoff is the best.

So my plan is to obtain all of the possible estimation values of a and b by doing combination method or any kind of methods. After obtaining so many pairs of a and b, I am not considering to simulate runoff and to validate it one by one by using each pair of a and b.

Thus the next step is I need to think of another method to get a single value of a and b. For example, if I have 100 pairs of a and b, then I can plot a and b separately using those 100 values. Maybe I can plot the Gaussian distribution then I can obtain the mean value of a and b. But I am not sure. I dont know how the plotting result will look like.

I am still thinking what to do next after obtaining many pairs of a and b in order to get the best single values of a and b. So I am looking forward to hear further comments from you. Thanks for your attention.

Best Regards, Intan

You really need to provide a (smallish) example that fully captures the problem at hand. From the current nb I cannot tell how your 'n' arises in the problem.