Calculate P(X(t)=x), the probability that a randomly selected customer makes exactly x transactions in the interval (0, t].
pnbd_nocov_PMF(r, alpha_0, s, beta_0, x, vT_i)
pnbd_staticcov_PMF(r, s, x, vAlpha_i, vBeta_i, vT_i)shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity of the purchase process
rate parameter of the Gamma distribution of the purchase process
shape parameter of the Gamma distribution for the lifetime process. The smaller s, the stronger the heterogeneity of customer lifetimes
rate parameter for the Gamma distribution for the lifetime process.
The number of transactions to calculate the probability for (unsigned integer).
Number of periods since the customer came alive.
Vector of individual parameters alpha.
Vector of individual parameters beta.
Returns a vector of probabilities.
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Bachmann P, Meierer M, Naef, J (2021). “The Role of Time-Varying Contextual Factors in Latent Attrition Models for Customer Base Analysis” Marketing Science 40(4). 783-809.
Fader PS, Hardie BGS (2005). “A Note on Deriving the Pareto/NBD Model and Related Expressions.” URL http://www.brucehardie.com/notes/009/pareto_nbd_derivations_2005-11-05.pdf.
Fader PS, Hardie BGS (2007). “Incorporating time-invariant covariates into the Pareto/NBD and BG/NBD models.” URL http://www.brucehardie.com/notes/019/time_invariant_covariates.pdf.
Fader PS, Hardie BGS (2020). “Deriving an Expression for P(X(t)=x) Under the Pareto/NBD Model.” URL https://www.brucehardie.com/notes/012/pareto_NBD_pmf_derivation_rev.pdf