Hi everyone,
I recently submitted my manuscript into a journal and got the following reply from a reviewer:
In section 4.1, it says: "The final result from MCS can be fitted to normal distribution as a consequence of the central limit theorem." I don't think this is true, since the solution given by MCS doesn't need to be normal distribution. Please justify this.
So, basically, I ran 10000 MC simulations in my simulation and fitted it to normal distribution with the understanding that the output follows normal distribution as the sampling number is pretty high. However, I saw some other authors fitting the final output to non-normal distribution too.
So, can the experts help me to please clarify this points to me and also suggest how I can suitably reply to the reviewer?
Thank you in advance
The mean of any distribution obeys the CLT and you can use it as a reasonable proxy to an estimate of precision for your mean, but it’s otherwise meaningless. There‘s no reason the assume the output of you MCS conforms to any particular distribution, it may even have multiple peaks and troughs or be discontinuous. Bootstrapping offers a much better estimate of precision from an MCS, by the way, than the CLT.
Hi,
You have to read carefully the Central limit theorem, because what you claim is not exactly what the theorem says.
The MC simulation out puts need not always be Gaussian. It depends on how well the model describe the data and how well the error co-variance matrix is measured. I do not quite understand your motivation to fit the MC simulation out puts to a normal distribution. In case of fitting a data, people just report the histogram of the MCMC outputs.
Niladri rightly says that the output of MC need not always be Gaussian. It depends on the characteristic of a model and its input(s). One can fit any data to the normal distribution but it may not give the meaningful conclusions in all cases.
As far as reviewer comments are concerned, it is not always necessary to agree with them and you can justify your approach with proper explanation.
Best Regards
"The final result from MCS can be fitted to normal distribution as a consequence of the central limit theorem".
This doesn't mean anything new. Just providing a justification for MCS which is based on the CLT.
The mean of any distribution obeys the CLT and you can use it as a reasonable proxy to an estimate of precision for your mean, but it’s otherwise meaningless. There‘s no reason the assume the output of you MCS conforms to any particular distribution, it may even have multiple peaks and troughs or be discontinuous. Bootstrapping offers a much better estimate of precision from an MCS, by the way, than the CLT.
A Monte Calo Method is based in the Law of Large Numbers, because basically you rely on repeating random experiments and then taking the average of them. The way Central Limit Theorem accounts in Monte Carlo Methods is the rate of convergence you can provide. So, the results of a Monte Carlo Simulation can fit almost everything; you can calculate the area under a function by MC, or the result can be just a number, like the price of a derivative, in case of financial applications. You can even estimate the value of a constant like \pi applying MC to the Buffon's needle problem for example.
Samundra,
You say "I ran 10000 MC simulations in my simulation and fitted it to normal distribution"
The real question is what model did you simulate? If you simulate a million Gamma, the resulting distribution will be Gamma. If you simulate 10 million Pareto, the resulting distribution will be Pareto. The Central Limit Theorem says nothing about those cases. It only applies if you look at means of a large number of identically distributed values.
If your paper can stand without this assumption of normality that you are making, seemingly without justification, then remove it and focus on the key results. If, on the other hand, normality of the simulated data is key but you have no other reason to justify it is normal, I would reconsider my methodology.
Good luck.
An unbiased estimator (UBE) of any quantity you wish to obtain from a Monte Carlo simulation converges to a Normal distribution around that quantity. That's essentially the what the Central Limit theorem says. This confuses people. If, for example, your model contains an exponentially distributed process, and you want to obtain the process time, you will end up in an exponential distribution which is built from a set of UBEs, each estimating the value of a bin of the obtained distribution. The estimation of each bin's value converges to a Normal distribution. Thus, following what Rees said, there's no reason to fit the UBE's mean to a Normal distribution, unless your research is about MC method. You just have to care about the fidelity of the model you're simulating, and that you run sufficiently large number of simulations.
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