Negative Binomial Approximations

We are working on an optimization problem that requires the

calculation of Negative Binomial probabilities and cumulatives. Programming

exact calcultions is not difficult until the mean and/or variance get too

high so that the machine underflows. We have been using a Camp-Paulsen type

approximation to the Neg Bi that comes from a paper of Bartko (Biometrics 8

pp340 - 342) as reported in Jonson & Kotz's book on Discrete Distributions.

In fact we have been using this approximation successfully for over a decade

but recently we ran into a problem with it's accuracy. The problem cases

are typically one with small means and huge variances (VMR of ~ 265). While

I had known about this problem before, these cases had not arisen with any

frequency until recently when the optimization problem was applied to a

whole new set of data.

My question: Is there any new work in this area. It seems to me

there are two ways to go. One is to develop an even better approximation

that the Camp-Paulsen although Bartko reports and I have observed mostly

excellen results. The other approach is to always use exact (recursive)

calculations and do good stuff to avoid underflow and the problems inherent

with summing a lot of numbers which differ in orders of magnitude.

Any help or pointers will be appreciated. Anyone interested in the

magnitude of the error might be interested in the following example:

Mean= 2.8 Variance = 775.33

Exact Pr{0}=.94.....

CP Pr{0} = .999......

In fact, exact PR{<=413}=.999..... This caused a huge differenc since the

optimality condition is based on cumulatives. When searching for a

cumulative of .999 or more, exact gave asolution of 413 while approximate

gave a solution of 0. Very ungood.

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