Background and Motivation: I was reading the paper RFM and CLV: Using Iso-Value Curves for Customer Base Analysis by Peter S. Fader, Bruce G. S. Hardie and Ka Lok Lee, in an attempt to gain some intuition behind the methods available in software libraries such as lifetimes and BTYD. The paper presents a way to incorporate RFM models in customer lifetime value calculations by using the gamma-gamma model for spend per transaction.

Problem: In the paper, the following formula, to calculate the expected average transaction value for a customer with an average spend of $m_x$ across $x$ transactions, is presented:

$$ \begin{align} \mathbb{E}(M\mid p, q, \gamma, m_x, x) & = \frac{(\gamma + m_xx)p}{px+q-1}\\ & = (\frac{q-1}{px+q-1})\frac{\gamma p}{q-1}+(\frac{px}{px+q-1})m_x\\ \end{align} $$

I understand that the $q$ parameter is used as the shape parameter in the underlying gamma distribution that models the heterogeneity in mean transaction values across customers, but I'm unable to grasp its effect on the expected average transaction value for a given customer.

Namely, I was able to fit a gamma-gamma model with $q < 1$ and the expected average transaction value for certain customers evaluates to a negative value.

Question: Does the fact that I get negative values for expected transaction values expose some issues with the underlying dataset used to fit the model?

Do models with $q < 1$ make sense, and if so, what should I interpret from a negative expected average transaction value?


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