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
)
```

- 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.

Returns a vector with the DERT for each customer.

`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.

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