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The saddlepoint approximation for a general birth process

Published online by Cambridge University Press:  14 July 2016

H. E. Daniels
H. E. Daniels*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge, CB2 1SB, U.K.
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Abstract

Saddlepoint approximations to the probabilities for a general time-homogeneous birth process are derived from the known Laplace transform with respect to time. Numerical computations for the simple (logistic) epidemic show the approximation to have a remarkably low relative error over the useful range of times for populations as small as 5. The merit or otherwise of renormalising the approximate probabilities is discussed. To study the accuracy for moderately large population sizes, a special form of birth rate is used, for which the exact probabilities can be computed directly.

Keywords

BIRTH PROCESSSIMPLE EPIDEMICSADDLEPOINT APPROXIMATION
Type
Research Paper
Information
Journal of Applied Probability , Volume 19 , Issue 1 , March 1982 , pp. 20 - 28
Copyright
Copyright © Applied Probability Trust 1982 

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References

Bailey, N. T. J. (1957) The Mathematical Theory of Epidemics. Griffin, London.Google Scholar
Bailey, N. T. J. (1968) A perturbation approximation to the simple epidemic in a large population. Biometrika 55, 199–209.Google Scholar
Barbour, A. D. (1972) The principle of the diffusion of arbitrary constants. J. Appl. Prob. 9, 519–541.Google Scholar
Bartholomew, D. J. (1973) Stochastic Models for Social Processes, 2nd edn. Wiley, London.Google Scholar
Billard, L. (1973) Factorial moments and probabilities for the general stochastic epidemic. J. Appl. Prob. 10, 277–288.Google Scholar
Daniels, H. E. (1954) Saddlepoint approximations in statistics. Ann. Math. Statist. 25, 631–650.Google Scholar
Daniels, H. E. (1960) Approximate solutions of Green's type for univariate stochastic processes. J.R. Statist. Soc. B 22, 376–401.Google Scholar
Daniels, H. E. (1979) Exact saddlepoint approximations. Biometrika 67, 59–63.Google Scholar
Mcneil, D. R. (1972) On the simple stochastic epidemic. Biometrika 59, 494–497.Google Scholar
Olver, F. W. J. (1974) Asymptotics and Special Functions. Academic Press, New York.Google Scholar
Severo, N. C. (1967) Two theorems on solutions of differential-difference equations and applications to epidemic theory. J. Appl. Prob. 4, 271–280.Google Scholar
Severo, N. C. (1969) Generation of some stochastic epidemic models. Math. Biosci. 4, 395–402.CrossRefGoogle Scholar
Taga, Y. and Isii, K. (1959) On a stochastic model concerning the pattern of communication-diffusion of news in a social group. Ann. Inst. Statist. Math. 11, 25–43.Google Scholar