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eppmsadga |
EPPM Gamma
Saddlepoint |

**DESCRIPTION**- Computes a saddlepoint approximation based on the gamma distribution to the probabilities for an Extended Poisson Process Model.
**USAGE**`eppmsadga(lambda)`**REQUIRED ARGUMENTS**`lambda`vector of positive birth rates. Missing values (NAs) are allowed but will usually produce an NA result. **OPTIONAL ARGUMENTS**`second`Logical variable. If `second=T`the second term correction to the saddlepoint approximation is included.**VALUE**- Numerical value giving the log-probability that N = n -1 where n = length(lambda).
**DETAILS**- 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 with the gamma distribution as leading term. The probabilities are exact when the lambda's are constant and the probabilities are Poisson.

The computation of probabilities for the pure birth process is central to extended Poisson process models for modelling count data. **REFERENCES**- 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.
**SEE ALSO**- eppmsadno, eppmsadnb, eppmsadzc, S-Plus programs for EPPM by Heather Podlich.

**EXAMPLES**`# Probability that N=3 for Poisson with mean 5``# Same as dpois(3,mean=5)`

> exp(eppmsadga(c(5,5,5,5)))

[1] 0.1403739

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Gordon Smyth.
Copyright © 1996-2016. *Last modified:
10 February 2004*