/ Home

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

 

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