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eppmsadzc EPPM 6 Cumulant Saddlepoint
Computes a saddlepoint approximation to the probabilities for an Extended Poisson Process Model.
lambda vector of positive birth rates. Missing values (NAs) are allowed but will usually produce an NA result.
Numerical value giving the log-probability that N = n -1 where n = length(lambda).
The function computes the log-probability mass for the count distribution resulting from a pure birth process at unit time. The waiting time until the next birth is exponential with mean lambda[n], where n is the number of births so far. Let N be the number of births at unit time. The probability that N = n depends on lambda[0:n]. The function takes the input vector to be lambda[0:n] and computes log P(N=n).

The computation uses a saddlepoint approximation based on matching the first 6 cumulants of the tilted distribution. An cumulant generating function is inverted by Gaussian quadrature. The accuracy of the saddlepoint is similar in most cases to that of the normal saddlepoint with second term correction.

The computation of probabilities for the pure birth process is central to extended Poisson process models for modelling count data.
Smyth, G. K., and Podlich, H. M. (2002). An improved saddlepoint approximation based on the negative binomial distribution for the general birth process. Computational Statistics 17, 17-28. [PDF]
Podlich, H. M., Faddy, M. J., and Smyth, G. K. (1999). Semi-parametric extended Poisson process models.
eppmsadnb, eppmsadno, S-Plus programs for EPPM by Heather Podlich.
# Exact value is actually 0.368
> exp(eppmsadzc(c(100,1,100,1)))
[1] 0.3579903
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Gordon Smyth. Copyright © 1996-2016. Last modified: 10 February 2004