Pareto/NBD: Discounted Expected Residual Transactions

Calculates the discounted expected residual transactions.

pnbd_nocov_DERT

Discounted expected residual transactions for the Pareto/NBD model without covariates

pnbd_staticcov_DERT

Discounted expected residual transactions for the Pareto/NBD model with static covariates

pnbd_nocov_DERT(
  r,
  alpha_0,
  s,
  beta_0,
  continuous_discount_factor,
  vX,
  vT_x,
  vT_cal
)

pnbd_staticcov_DERT(
  r,
  alpha_0,
  s,
  beta_0,
  continuous_discount_factor,
  vX,
  vT_x,
  vT_cal,
  mCov_life,
  mCov_trans,
  vCovParams_life,
  vCovParams_trans
)

Arguments

r

shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity of the purchase process

alpha_0

rate parameter of the Gamma distribution of the purchase process

s

shape parameter of the Gamma distribution for the lifetime process. The smaller s, the stronger the heterogeneity of customer lifetimes

beta_0

rate parameter for the Gamma distribution for the lifetime process.

continuous_discount_factor

continuous discount factor to use

vX

Frequency vector of length n counting the numbers of purchases.

vT_x

Recency vector of length n.

vT_cal

Vector of length n indicating the total number of periods of observation.

mCov_life

Matrix containing the covariates data affecting the lifetime process. One column for each covariate.

mCov_trans

Matrix containing the covariates data affecting the transaction process. One column for each covariate.

vCovParams_life

Vector of estimated parameters for the lifetime covariates.

vCovParams_trans

Vector of estimated parameters for the transaction covariates.

Value

Returns a vector with the DERT for each customer.

Details

mCov_trans is a matrix containing the covariates data of the time-invariant covariates that affect the transaction process. Each column represents a different covariate. For every column a gamma parameter needs to added to vCovParams_trans at the respective position.

mCov_life is a matrix containing the covariates data of the time-invariant covariates that affect the lifetime process. Each column represents a different covariate. For every column a gamma parameter needs to added to vCovParams_life at the respective position.

References

Schmittlein DC, Morrison DG, Colombo R (1987). “Counting Your Customers: Who-Are They and What Will They Do Next?” Management Science, 33(1), 1-24.

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