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eppmsadno |
EPPM Saddlepoint
Approximation |

**DESCRIPTION**- Computes the saddlepoint approximation to the probabilities for an Extended Poisson Process Model described by Daniels (1982).
**USAGE**`eppmsadno(lambda, second=T)`**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 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 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. **REFERENCES**- 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] **SEE ALSO**- eppmsadnb, eppmsadga, eppmsadzc, eppminvmgf, S-Plus programs for EPPM by Heather Podlich.

**EXAMPLES**`# 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*