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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

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