Negative Binomial Approximations
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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
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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Sat, 19 Oct 1996 22:40:47 GMT

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