- Computes the saddlepoint approximation to the probabilities for an Extended Poisson
Process Model described by Daniels (1982).
- eppmsadno(lambda, second=T)
- REQUIRED ARGUMENTS
||vector of positive birth rates. Missing values (NAs) are allowed but will usually
produce an NA result.
- OPTIONAL ARGUMENTS
||Logical variable. If second=T the second term correction to the saddlepoint
- 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.
- SEE ALSO
- 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
# Worst case - one lambda is much smaller than the others
# The exact value is 0.9049279
> exp( eppmsadno(c(1000,0.1)) )
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10 February 2004