Is it the model or the distributions?

In a previous post I made a brief (but probably controversial to some) statement. I stated that often the type of input distributions used is not that important because the nature of the model has a much more significant affect upon the shape of the output distribution. I therefore stated that when dealing with the input distribution you probably should not worry too much about the moments other than the mean or the standard deviation for input variables.

 This is a fairly bold statement, and it probably requires more support. So why do I believe that you often don’t need to worry too much about the distribution type? This conviction is a result of what I experienced during my PhD. During my PhD I had to model a number of manufacturing operations. This was to ascertain heuristics that could be used to allocate a distribution to variable once the associated method of manufacture was known. I frequently found that it was the phenomena underlying the manufacturing method (and thus the model of that operation) that had the greatest affect upon the calculated distribution. This was because the central limit theorem would often be present because multiplicative and additive operations are so common. In fact, it is almost impossible to develop a model of any real system that does not include a number of additive or multiplicative operations.

This is certainly not always the case. I found that in cases where there a small number of random variables there was a reduced opportunity for the central limit theorem to express itself, and the type of distribution became very important. However, the majority of cases considered in the real world have a large number of random variables. Therefore, the central limit theorem is more common, and the distribution type is less important in the majority of real life cases. Still, be aware of cases where this might not be the case, especially when there are only a small number of random variables.

If you wish to verify this contention of mine, consider the models that you often deal with. You will most likely find that there are a number of additive and multiplicative functions within any of these models. You can also try changing the distribution types to check that they do not matter that much. Of course, keep the mean and the standard deviation the same.