Pareto/NBD Model

The Pareto/NBD model is the foundational buy-till-you-die model of customer-base analysis, answering the question of which customers are still active and how many transactions they will make in the future from a non-contractual purchase history. Introduced by David Schmittlein, Donald Morrison and Richard Colombo in their 1987 Management Science paper "Counting Your Customers," it combines two stochastic stories: customers buy according to a Poisson process while alive, and each customer has an unobserved lifetime after which they are permanently inactive. Purchasing rates vary across customers by a gamma distribution, producing the negative binomial (NBD) for counts, and dropout rates also vary by a gamma distribution, producing a Pareto distribution of lifetimes, which gives the model its name. Unlike later discrete-dropout variants, the Pareto/NBD allows a customer to become inactive at any instant in continuous time, not only after a purchase. From only each customer's recency, frequency and tenure, the model yields a probability that the customer is still alive and an expectation of their future buying. Its main cost is computational: estimation involves Gaussian hypergeometric functions and careful numerical integration, which historically made it hard to apply.