# Does it make sense to fit a Pareto Curve to sales data?

The Pareto principle is used surprisingly widely in business. I'm wondering how correct it's wide use is. It seems as though it is often used without empirical verification as if the situation fits a Pareto distribution a priori.

Here's an example:

Let's say there is a retail store with a wide range of items in a single aisle. The items are assumed to be a Pareto distribution. The sales data of the items are then fit to a Pareto curve and shown to fit it (using R^2, a basic measurement). This curve would in turn help you find out, I suppose, what the best items would be to take out and so on.

My question is two-fold:

1) Is it actually so easy to empirically verify whether the aisle's historical sales is a Pareto distribution? 2) Does fitting the data to a Pareto curve mean anything? I've seen that, broadly, fitting data to a Pareto curve does not necessarily imply that the data is actually Pareto distributed.

• Shalizi has some fun blog posts on fitting power law distributions, e.g. this one. Commented Jun 10, 2014 at 13:27
• When you say you're "using $R^2$", could you explain how you're using it? I can think of suitable and unsuitable uses for $R^2$ in such an exercise, depending on what you're looking at the $R^2$ of. Commented Jun 10, 2014 at 13:30
• @Glen_b The R^2 of the fitted Pareto curve to the data Commented Jun 10, 2014 at 13:33
• How is it being computed? Commented Jun 10, 2014 at 13:37
• (+1) It is refreshing to see healthy scepticism about a statistical procedure expressed so thoughtfully. Welcome to our site!
– whuber
Commented Jun 10, 2014 at 13:38

Sometimes the Pareto is used to represent an exponential with gamma mixture. This is especially common in corporate statistics such as marketing and risk management. In other words people (or items in this case) do something or other according to an exponential distribution (in this case get sold) but because items differ from each other the rate parameter of the exponential differs and therefore the statistician uses a gamma distribution (because gamma has many shapes) to represent those differing rate parameters.

This is explained in this blog post but reproduced here for longevity. To illustrate consider the exponential that is intended to represents when an item will sell:

$$f(x)= \theta e^{-\theta x}$$

This pdf can described as a conditional pdf just by changing the notation like this.

$$f_{X\mid\Theta}(x\mid\theta)= \theta e^{-\theta x}$$

If we grant that $$\theta$$ itself is distributed according to a gamma distribution with scale $$\alpha$$ and shape $$\beta$$ then the pdf of $$\theta$$ looks like this:

$$f_\Theta(\theta)=\frac{\alpha^{\beta}}{\Gamma(\beta)}\theta^{\beta-1}e^{-\alpha \theta}$$

If you are not familiar with the gamma distribution, $$\Gamma(x)$$ is the gamma function.

Constructing the unconditional exponential from the conditional plus the gamma means we have a 'density' of weights assigned to each $$\theta$$ in the exponential.

$$f_X(x) = \int^\infty_0 = f_{X\mid \Theta}(x\mid\theta) f_\Theta(\theta) \, d\theta$$

If you compute this integral (which I will leave as an exercise worth doing) you will get the Pareto defined by:

$$f_X(x)= \frac{\beta \alpha^\beta}{(x+\alpha)^{\beta+1}}$$

So the following two assumptions worth questioning are made.

1. items are sold according to an exponential rate.
2. differences in the rate of sales of items is gamma.

Interesting note: For overdispersed poisson the negative binomial distribution (NBD) is often used. Often this overdispersion is attributed to heterogeneity in whatever is being modeled with a poisson. The NBD is therefore used because it is a similar mixture distribution also known as the gamma-poisson where the gamma is the distribution over the poisson's $$\lambda$$. The assumptions are very similar.

1) Is it actually so easy to empirically verify whether the aisle's historical sales is a Pareto distribution?

You cannot identify that sample data come from any specific distribution. After all, even the tiniest deviation from that distributional form would make it from some other distribution.

You can, however, often identify that data isn't from some distribution, since many distributions will be extremely unlikely to have generated some samples, sometimes so unlikely (compared to alternatives) that one is prepared to conclude that it's not the case.

Does fitting the data to a Pareto curve mean anything?

Sure, it tells you the best-fitting Pareto.

I've seen that, broadly, fitting data to a Pareto curve does not necessarily imply that the data is actually Pareto distributed.

Correct. Indeed, even without looking at any set of data, you can say it almost certainly isn't exactly Pareto.

However, you can see whether a Pareto is a plausible model for the data, and it's possible to explore how much the kind of deviations the data might have from being Pareto might impact your inferences/conclusions.

To respond to the question in your title, I think it can make sense, as long as one doesn't make the mistake of believing your model is actually true.