- Computes a saddlepoint approximation to the probabilities for an Extended Poisson
- 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
approximation is 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 negative binomial
distribution. The probabilities are exact whenever the lambda can be sorted to form an
arithmetic increasing sequence. Amongst other things, this means that eppmsadnb
will compute binomial, negative binomial or Poisson probabilities exactly including the
case where the sizes are not integers. In the worst cases, 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.
- 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.
- Podlich, H. M., Faddy, M. J., and Smyth, G. K. (1999). Semi-parametric extended Poisson
- SEE ALSO
- eppmsadno, eppmsadzc, S-Plus programs for EPPM by Heather Podlich.
- # Probability that N=3 for Poisson with mean 5
- # Same as dpois(3,mean=5)
# Probability that N=3 for negative binomial
- # with size=2 successes and success probablity exp(-1)
# Same as dnbinom(3,size=2,prob=exp(-1))
# Probability that N=3 for binomial distribution
# with 5 trials and success probability 1-exp(-1)
- # Same as dbinom(3,size=5,prob=1-exp(-1))
# Worst case - the lambda form two clusters at very different values
# Exact value is actually 0.368
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10 February 2004