bgbb.pmf.General {BTYD}

BG/BB General Probability Mass Function

Description

Calculates the probability that a customer will make x.star transactions in the first n.star transaction opportunities following the calibration period.

Usage

bgbb.pmf.General(params, n.cal, n.star, x.star)

Arguments

params

BG/BB parameters - a vector with alpha, beta, gamma, and delta, in that order. Alpha and beta are unobserved parameters for the beta-Bernoulli transaction process. Gamma and delta are unobserved parameters for the beta-geometric dropout process.

n.cal

number of transaction opportunities in the calibration period. Can also be a vector of calibration period transaction opportunities - see details.

n.star

number of transaction opportunities in the holdout period, or a vector of holdout period transaction opportunities.

x.star

number of transactions in the holdout period, or a vector of transaction frequencies.

Details

P(X(n, n + n*) = x* | alpha, beta, gamma, delta). This is a more generalized version of the bgbb.pmf. Setting n.cal to 0 reduces this function to the probability mass function in its usual format - the probability that a user will make x.star transactions in the first n.star transaction opportunities.

It is impossible for a customer to make a negative number of transactions, or to make more transactions than there are transaction opportunities. This function will throw an error if such inputs are provided.

n.cal, n.star, and x.star may be vectors. The standard rules for vector operations apply - if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be a vector of probabilities.

Value

Probability of X(n, n + n*) = x*, given BG/BB model parameters.

References

Fader, Peter S., Bruce G.S. Hardie, and Jen Shang. "Customer-Base Analysis in a Discrete-Time Noncontractual Setting." Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. Web.

See Also

bgbb.pmf

Examples

params <- c(1.20, 0.75, 0.66, 2.78)
# Probability that a customer will make 3 transactions in the 10
# transaction opportunities following the 6 transaction opportunities
# in the calibration period, given BG/BB parameters.
bgbb.pmf.General(params, n.cal=6, n.star=10, x.star=3)

# Vectors may also be provided as input:
# Comparison between different frequencies:
bgbb.pmf.General(params, n.cal=6, n.star=10, x.star=1:10)
# Comparison between different holdout transaction opportunities:
bgbb.pmf.General(params, n.cal=6, n.star=5:15, x.star=3)