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eppmsadno EPPM Saddlepoint Approximation
Computes the saddlepoint approximation to the probabilities for an Extended Poisson Process Model described by Daniels (1982).
eppmsadno(lambda, second=T)
lambda vector of positive birth rates. Missing values (NAs) are allowed but will usually produce an NA result.
second Logical variable. If second=T the second term correction to the saddlepoint approximation included.
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 the normal distribution, as described by Daniels (1982). The second term correction is included, unless second=F. The worst case occures when one lambda[n] is much smaller than the others; then the probabilities are accurate to 2 significant figures.

The computation of probabilities for the pure birth process is central to extended Poisson process models for modelling count data.
Daniels, H. E. (1982). The saddlepoint approximation for a general birth process. Journal of Applied Probability, 19, 20-28.
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]
eppmsadnb, eppmsadga, eppmsadzc, eppminvmgf, S-Plus programs for EPPM by Heather Podlich.
# Probability that N=3 for Poisson with mean 5
# The exact value 0.1403739
> exp(eppmsadno(c(5,5,5,5)))
[1] 0.140337

# Worst case - one lambda is much smaller than the others
# The exact value is 0.9049279
> exp( eppmsadno(c(1000,0.1)) )
[1] 0.8995597
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Gordon Smyth. Copyright © 1996-2016. Last modified: 10 February 2004