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An Improved Saddlepoint Approximation Based on the Negative Binomial Distribution for the General Birth Process

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Summary

Daniels (JAP 1982) gave a saddlepoint approximation to the probabilities of a general birth process. This paper gives an improved approximation which is only slightly more complex than Daniels’ approximation and which has considerably reduced relative error in most cases. The new approximation has the characteristic that it is exact whenever the birth rates can be reordered into a linear increasing sequence.

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Acknowledgement

Much of the work leading to this paper was completed while the first author was a visiting Associate Professor at the University of Southern Denmark.

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Appendix

Appendix

Suppose that λ0, …, λn, n ≥ 1, are non-negative and that \(\sum\nolimits_{i = 0}^n {1{\rm{/}}\left( {{\lambda _i} - \widetilde\theta } \right) = 1} \). Let \({\kappa _2} = \sum\nolimits_{i = 0}^n {1/{{\left( {{\lambda _i} - \widetilde\theta } \right)}^2}} \). Then 1/(n + 1) ≤ κ2 ≤ 1. We seek λ and a such that \(\sum\nolimits_{i = 0}^n {1/\left( {{\lambda _i} + ai} \right)} = 1\) and \(\sum\nolimits_{i = 0}^n {1/{{\left( {\lambda + ai} \right)}^2}} = {\kappa _2}\). If the λi; are all equal then the solution is a = 0 and λ = λ0. If the λi are not all equal then a ≠ 0 and b = λ/a is finite. Then a can be solved in terms of b as

$$a = \sum\limits_{i = 0}^n {1/\left( {b + i} \right)} $$

and b solves

$$\frac{\sum\nolimits_{i=1}^{n} 1 /(b+i)^{2}}{\left\{\sum\nolimits_{i=0}^{n} 1 /(b+i)\right\}^{2}}=\kappa_{2}.$$
((4))

The left-hand side of (4) tends to 1 as b → 0 and to 1/(n + 1) as b → ∞ and is monotonic decreasing between these two values. There is therefore a unique non-negative solution for b and hence for a. The left-hand of (4) is in fact convex in b so that Newton’s method for finding b is monotonically convergent from a suitable starting value.

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Smyth, G.K., Podlich, H.M. An Improved Saddlepoint Approximation Based on the Negative Binomial Distribution for the General Birth Process. Computational Statistics 17, 17–28 (2002). https://doi.org/10.1007/s001800200088

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  • DOI: https://doi.org/10.1007/s001800200088

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