Generalized Pareto/NBD Probability Mass Function

Description

Generalized probability mass function for the Pareto/NBD.

Usage

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pnbd.pmf.General(params, t.start, t.end, x)

Arguments

params

Pareto/NBD parameters - a vector with r, alpha, s, and beta, in that order. r and alpha are unobserved parameters for the NBD transaction process. s and beta are unobserved parameters for the Pareto (exponential gamma) dropout process.

t.start

start of time period for which probability is being calculated. It can also be a vector of values.

t.end

end of time period for which probability is being calculated. It can also be a vector of values.

x

number of repeat transactions by a random customer in the period defined by (t.start, t.end]. It can also be a vector of values.

Details

P(X(t.start, t.end)=x | r, alpha, s, beta). Returns the probability that a customer makes x repeat transactions in the time interval (t.start, t.end].

It is impossible for a customer to make a negative number of repeat transactions. This function will return an error if it is given negative times or a negative number of repeat transactions. This function will also return an error if t.end is less than t.start.

t.start, t.end, and x 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 transaction occuring between t.start and t.end conditional on model parameters. If t.start, t.end, and/or x has a length greater than one, a vector of probabilities will be returned.

References

Fader, Peter S., and Bruce G.S. Hardie. “Deriving an Expression for P (X(t) = x) Under the Pareto/NBD Model.” Sept. 2006. Web. http://www.brucehardie.com/notes/012/

Fader, Peter S., Bruce G.S. Hardie, and Kinshuk Jerath. “Deriving an Expression for P (X(t, t + tau) = x) Under the Pareto/NBD Model.” Sept. 2006. Web. http://www.brucehardie.com/notes/013/

Examples

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params <- c(0.55, 10.56, 0.61, 11.64)
# probability that a customer will make 10 repeat transactions in the
# time interval (1,2]
pnbd.pmf.General(params, t.start=1, t.end=2, x=10)
# probability that a customer will make no repeat transactions in the
# time interval (39,78]
pnbd.pmf.General(params, t.start=39, t.end=78, x=0)

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