Bayesian Analysis For Customer Lifetime Model Under Noncontractual Setting

The noncontractual setting of customer-base analysis is one of the hot topics in marketing science and management science.The main objective of the executives(or researchers)is tracking and managing customer lifetime value systematically by constructing and analyzing model to concern about issues such as customer churn,attrition,retention and customer lifetime value.The existing models for noncontractual setting of customer-base analysis are developed by combining purchasing model(counting model)with lifetime model(timing model).A standard purchasing model that has been widely used is negative binomial distribution(NBD).However,there is no uniform and standard model for lifetime model,one of the most important reason is that when modeling the lifetime,there are different explanations for the event about how/when customers become inactive,in other words,different explanation leads to different customer lifetime model.In addition,the existing methods developed for almost all customer lifetime models often assumed that the distribution of the heterogeneity parameters is a standard parametric distribution,this may lead to unreasonable statistical inference.Therefore,this dissertation develop a standard customer lifetime model for the noncontractual setting of customer-base analysis from the perspective of survival analysis,that is G/W customer lifetime model,and combine this model with standard purchasing model to deriving a series of properties of the G/W/NBD model at the individual-level and the aggregate-level,respectively.Then,for the existing customer lifetime models,this dissertation develop a nonparametric hierarchical customer lifetime model by assuming that the heterogeneity parameters in these models come from some unknown mixtion distribution,and putting a Dirichlet process prior on the mixing distribution,using nonparametric empirical Bayes method to analysis the nonparametric hierarchical customer lifetime model.Finally,under the assumption that the heterogeneity parameters in purchase model and customer lifetime model come from some unknown mixtion distribution,this dissertation develop a nonparametric hierarchical purchase model and customer lifetime model to modeling the purchase event and duration time simultaneously in the noncontractual setting of customer-base analysis,in which unknown distributions of heterogeneity parameters are approximated by a truncated Dirichlet process,and a nonparametric empirical Bayesian method is developed to obtain Bayesian estimations of unknown parameters in the proposed nonparametric models.The main purposes of this dissertation include:1.Motivated by the fact that the time when a customer becomes inactive can be treated as a "time to death" event and the Weibull distribution can flexibly capture the time event,we propose a new model,i.e.,the G/W model,by using the Weibull distribution to model "time to death" event.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 t ime p eriod.S everal s imulation s tudies a re c onducted t o i nvestigate the forecasting accuracy and flexibility o f the proposed m odel.A C DNOW d ata set is analyzed by the proposed model,and compare its performance with Pareto/NBD model and G/G/NBD model at several aspect.2.Based on the heterogeneity parameter in existing customer lifetime models often be assumed a standard parametric distribution,which may lead to bias due to the misspecification of parametric models,we relax the parametric assumption imposed on the heterogeneity parameter in customer lifetime model using the truncated Dirichlet process(DP)as a nonparametric prior for such heterogeneity parameter to propose a nonparametric customer lifetime model.Moreover,to address the issue that specify a prior distribution on hyper-parameters may has large subjectivity and arbitrariness,we develop a generalization Dirichlet process mixture customer lifetime model based on general mixture customer lifetime model by treating the mixing distribution as a infinite dimension p arameter,and putting a Dirichlet process prior on the mixing distribution without specifying the hyper-prior and hyper-parameters for the unknown Dirichlet process parameters.A nonparametric empirical Bayes approach is applied to analysis the generalization Dirichlet process mixture customer lifetime model.The stick-breaking prior and the blocked Gibbs sampler are used to sample the posterior samples form conditional posterior distribution efficiently.In simulation study,we consider three common customer lifetime model:Exponential distribution,Gompertz distribution and Weibull distribution.Extensive simulation studies and two real datasets are conducted to illustrate the flexibility and effectiveness of the proposed approach in approximating the distribution of heterogeneity parameter with diverse shape.3.In the noncontractual setting of customer-base analysis,heterogeneity parameters in purchase model and lifetime model are usually assumed to follow some familiar parametric distribution such as gamma or log-normal distribution.But,in many applications,these assumptions may be questionable because the true distributions of heterogeneity parameters are usually unknown.To this end,this paper relaxes these assumptions imposed on heterogeneity parameters to develop a nonparametric approach to purchase model and lifetime model,in which unknown distributions of heterogeneity parameters are approximated by a truncated Dirichlet process.A nonparametric empirical Bayesian method is developed to obtain Bayesian estimations of unknown parameters in the proposed nonparametric models.The blocked Gibbs sampler is presented to draw observations required for Bayesian inference from the corresponding posterior distributions of the components of parameters.Extensive simulation studies and a CDNOW data set are presented to illustrate the newly developed methodologies.