The Gamma/Weibull Customer Lifetime Model
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The Gamma/Weibull Customer Lifetime Model Gen Ye1 · Songjian Wang1
Received: 29 November 2017 / Revised: 20 March 2018 / Accepted: 28 April 2018 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper proposes a new customer lifetime model: the Gamma/Weibull distribution (G/W). Similar to the Pareto/NBD model, we propose a G/W/NBD model by combining the G/W distribution with a negative binomial distribution (NBD) and study its properties such as (i) the probability that a customer to be alive at a time point; (ii) the expectation and variance of the number of transactions for a customer during a fixed time period; (iii) the conditional expectation and conditional variance of the number of future transactions for a customer during a fixed time period. Several simulation studies are conducted to investigate the forecasting accuracy and flexibility of the proposed model. A CDNOW data set is analyzed by the proposed model. Keywords Customer lifetime · Gamma distribution · Negative binomial distribution · Purchase history · Weibull distribution Mathematics Subject Classification 62E15 · 62F10 · 62P05
1 Introduction The customer lifetime model has been widely studied in marketing science and management science over the past years. For example, Schmittlein et al. [1] developed a Pareto/NBD model to count and identify those customers who are still active based on the number and timing of the customer’s previous transactions. Although the Pareto/NBD model is useful, it has been proven to be a difficult model to implement
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Gen Ye [email protected] Key Lab of Statistical Modeling and Data Analysis of Yunnan Province, Yunnan University, Kunming 650091, People’s Republic of China
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G. Ye, S. Wang
because of computational challenges in relation to parameter estimation. To address the issue, Fader et al. [2] proposed a beta-geometric/NBD (BG/NBD) model, which represents a slight variation in the behavioral “story” associated with the Pareto/NBD, and showed that the parameters in the BG/NBD model can be easily evaluated in Microsoft Excel. Bemmaor and Glady [3] proposed a flexible customer lifetime model, i.e., the Gamma/Gompertz (G/G) model, and derived the moments of the distribution of the number of transactions over the fixed time periods (0, T ] and (T, T + T ∗ ] via the G/G/NBD model that was obtained by combining the G/G model with the NBD model. Their numerical results showed that the G/G/NBD model was slightly better than the Pareto/NBD model for forecasting the mean number of transactions. There are many applications and extensions for the Pareto/NBD model (e.g., Kumar [4], Jain and Singh [5], Reinartz and Kumar [6], Ho et al. [7], Fader et al. [8], Makoto [9]). However, from the perspective of survival analysis, customer lifetime can be regarded as a time to an event of interest, that is, “time to death”. The type of data is called event-time data or survival data. The existing models mentione
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