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Evidence · Brand growth

NBD-Dirichlet model: what the research shows

The NBD-Dirichlet model reproduces the main patterns of repeat buying, including penetration, purchase frequency, share of category requirements and duplication between brands, from each brand's market share and two category parameters. It has fitted data from more than 40 product fields and many countries, and its deviations from real data are systematic and informative. It assumes a stable, unsegmented market, so trending or partitioned markets need care or extensions.

Definition

The NBD-Dirichlet is a stochastic model of how often people buy a category and how they split those purchases between brands. It combines four assumptions: each consumer buys the category at a steady random (Poisson) rate; these rates vary across consumers following a Gamma distribution, which yields the negative binomial distribution (NBD) of purchase counts; each purchase is a random brand choice; and brand choice probabilities vary across consumers following a Dirichlet distribution. Its predictions serve as benchmarks for real brand performance.

What the research finds

It grew from one purchase-count regularity

The tradition began with the finding that the number of purchases of a brand per consumer in a period closely follows the negative binomial distribution. The full model, published in 1984, integrated many such regularities and was shown to describe buying in more than 40 product fields.[1][2]

Few inputs, many predictions

Given only each brand's sales and two category parameters, the model predicts penetration, purchase frequency, share of category requirements, sole buyers and repeat buying. Observed brand measures follow these lawlike patterns: large and small brands differ mainly in their number of buyers, and few buyers are fully loyal to one brand.[2][3][4]

It provides norms for judging brands

Because the model shows what is normal for a brand of a given size, analysts can compare actual performance against it to audit brands, anticipate how a new brand will perform and judge the effects of promotions or advertising. Double jeopardy itself is one of the model's predictions.[3][5]

Deviations are systematic

A replication across 62 packaged-goods categories found three kinds of deviation: small consistent gaps across all brands that point to non-stationary markets, persistent gaps for particular brands that point to market partitions, and gaps that reflect brands growing or declining. Buying propensities are never fully fixed or independent, yet brand performance stays close to prediction.[6]

Judging fit needs benchmarks

There has been no agreed way to decide when the model fits well enough. One study proposes six goodness-of-fit statistics, including two new ones, with benchmarks from 54 UK packaged-goods categories. A working paper proposes an alternative way to estimate the parameters that corrects a systematic error for large brands.[7][8]

Extensions and new settings

The model has been extended to include consumer characteristics such as demographics, so that they can explain differences in brand performance measures. Applied to two years of Italian sparkling wine purchases, it identified at least three groups of brands with different positions. A separate behavioural-economics tradition, set out in a book, reaches a similar picture of buyers spreading purchases across a small set of substitutable brands.[9][10][11]

How strong is the evidence?

The model's fit to stable repeat-purchase markets is supported by decades of studies across many categories and countries, and its core papers are peer-reviewed and highly cited. Much of the development and testing comes from researchers linked to one tradition, the Ehrenberg-Bass Institute and its predecessors. The model assumes stationary, unsegmented markets; trending markets, partitioned categories and subscription markets require extensions or separate parameters. It describes buying patterns rather than explaining why individual consumers choose, and one of the estimation improvements cited here is a working paper that has not been peer-reviewed.

Claims tested against the evidence

References

  1. A. S. C. Ehrenberg (1959). The Pattern of Consumer Purchases. Applied Statistics (Journal of the Royal Statistical Society, Series C).
  2. Gerald J. Goodhardt, Andrew S. C. Ehrenberg, Christopher Chatfield (1984). The Dirichlet: A Comprehensive Model of Buying Behaviour. Journal of the Royal Statistical Society. Series A (General).
  3. Andrew S. C. Ehrenberg, Mark D. Uncles, Gerald J. Goodhardt (2004). Understanding Brand Performance Measures: Using Dirichlet Benchmarks. Journal of Business Research.
  4. Mark D. Uncles, Andrew S. C. Ehrenberg, Kathy Hammond (1995). Patterns of Buyer Behavior: Regularities, Models, and Extensions. Marketing Science.
  5. Carl Driesener, Cam Rungie (2022). The Dirichlet Model in Marketing. Journal of Consumer Behaviour.
  6. John Scriven, John Bound, Charles Graham (2017). Making Sense of Common Dirichlet Deviations. Australasian Marketing Journal (AMJ).
  7. Carl Driesener, Melissa Banelis, Cam Rungie (2017). If the Model Fits, Use It: Methods and Benchmarks for Evaluating NBD-Dirichlet Goodness-of-Fit. Australasian Marketing Journal (AMJ).
  8. Nick Danenberg, Phouthun Nay (2024). New Approaches to Estimating NBD-Dirichlet Model Parameters, from Measuring Goodness-of-Fit using Euclidean Distance. SSRN Electronic Journal.
  9. Cam Rungie, Mark Uncles, Gilles Laurent (2013). Integrating Consumer Characteristics into the Stochastic Modelling of Purchase Loyalty. European Journal of Marketing.
  10. Francesca Bassi (2025). The NBD-Dirichlet model for Italian sparkling wines: understanding competitive relationships among brands. Statistical Methods & Applications.
  11. Gordon R. Foxall, Jorge M. Oliveira-Castro, Victoria K. James, Teresa C. Schrezenmaier (2007). The Behavioral Economics of Brand Choice. Palgrave Macmillan.

Bibliographic details only. Follow the DOI to read each study at its publisher.

Related topics

A synthesis of 11 studies, written by ScholarSignal. Last reviewed 2026-09-26.

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