# Can demand be modeled by a limited amount of probability distributions?

In stochastic inventory control, it is very important to model the demand appropriate to set e.g. order levels. Demand can be fit to a certain probability distribution via many means (e.g. a goodness of fit test). However, following common literature of stochastic inventory control (eg Axsäter, 2006; Silver, 1998), probability distributions are assigned to a set of demand points based on:

• Demand volume: high or low demand
• Variability: using the first two statistical moments to determine a coefficient that tells us something about the stability in demand patterns.

Demand in most practical situations is nonnegative integers and when you have relatively low demand, it is natural to use a discrete demand model. When the demand is relatively high, it is better to use a continuous one. But also within the discrete demand models, you can have multiple probability distributions, as well as within the continuous models. The choice of specific distributions within these classes (discrete and continuous) depend then on a certain variation coefficient. The discrete models that show up the most are the Poisson distribution and a compounding (logarithmic) distribution (i.e. a negative binomial distribution), where for the continuous models, the normal and gamma distribution show up the most.

Using this approach is far more practical and is often a good theoretical approach as well, as long as you use the right boundaries to separate volume & variability in demand patterns. At least it looks that way, when certain authorities in the field say it. But also a lot of software packages use this classification to assign distributions to model the demand.

Now, assuming you have an optimal threshold that distinct demand patterns based on demand volume and variability described by the first two statistical moments of the demand points, I have a strong feeling that the following and former mentioned four probability distributions (or a combination) are sufficient to approximate every demand distribution in practice with an acceptable accuracy:

• The pure Poisson distribution (relatively low demand & relatively stable (low variability) demand patterns)

• The negative binomial distribution (relatively low demand & relatively unstable (high variability) demand patterns)

• The normal distribution (relatively high demand & relatively stable (low variability) demand patterns)

• The gamma distribution (relatively high demand & relatively unstable (high variability) demand patterns)

This feeling I got is purely based on literature and my own experiences as a research scientist in the field. And with accuracy I mean: what is the accuracy of the CDF from random samples of one the four distributions (or a combination) against samples from the 'correct' distribution. Let's say an acceptable accuracy is $$95\%$$.

But now the question is: has any research been done on this and if yes, could someone provide me with information on this research that could evidence my feeling or reject it? Or just any comments, thoughts on this are more than welcome!

• What if you have an empiirical distribution which looks like none of those? Commented Jan 26, 2020 at 19:30
• Then the question is: how is the accuracy of using one the four (or a combination of them) compared to using the empirical (ie the right one) one? I'm mostly interested in the cdf-accuracy. Commented Jan 26, 2020 at 20:33