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The linear birth‒death process: an inferential retrospective

Part of: Parametric inference Inference from stochastic processes Applications

Published online by Cambridge University Press:  01 February 2019

Simon Tavaré
Simon Tavaré*
Affiliation:
University of Cambridge
*
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK. Email address: simon.tavare@cruk.cam.ac.uk
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Abstract

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In this paper we provide an introduction to statistical inference for the classical linear birth‒death process, focusing on computational aspects of the problem in the setting of discretely observed processes. The basic probabilistic properties are given in Section 2, focusing on computation of the transition functions. This is followed by a brief discussion of simulation methods in Section 3, and of frequentist methods in Section 4. Section 5 is devoted to Bayesian methods, from rejection sampling to Markov chain Monte Carlo and approximate Bayesian computation. In Section 6 we consider the time-inhomogeneous case. The paper ends with a brief discussion in Section 7.

Keywords

Approximate Bayesian computationMarkov chain Monte Carloestimation

MSC classification

Primary: 62M05: Markov processes
Secondary: 62F15: Bayesian inference 62P10: Applications to biology and medical sciences

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 253 - 269
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Abate, J. and Whitt, W. (1992).Numerical inversion of probability generating functions.Operat. Res. Lett. 12,245–251.Google Scholar
[2]Bailey, N. T. J. (1964).The Elements of Stochastic Processes with Applications to the Natural Sciences.John Wiley,New York.Google Scholar
[3]Branson, D. (1991).Inhomogeneous birth-death and birth-death-immigration processes and the logarithmic series distribution.Stoch. Process. Appl. 39,131–137.Google Scholar
[4]Brooks, S., Gelman, A., Jones, G. L. and Meng, X.-L. (eds) (2011).Handbook of Markov Chain Monte Carlo.CRC Press,Boca Raton, FL.Google Scholar
[5]Crawford, F. W. and Suchard, M. A. (2012).Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution.J. Math. Biol. 65,553–580.Google Scholar
[6]Crawford, F. W., Minin, V. N. and Suchard, M. A. (2014).Estimation for general birth-death processes.J. Amer. Statist. Assoc. 109,730–747.Google Scholar
[7]Davison, A. C., Hautphenne, S. and Kraus, A. (2018). Parameter estimation for discretely-observed linear birth-and-death processes. Preprint. Available at https://arxiv.org/abs/1802.05015v1.Google Scholar
[8]Fan, Y. and Sisson, S. A. (2018). ABC samplers. Preprint. Available at https://arxiv.org/abs/1802.09650v1.Google Scholar
[9]Feller, W. (1939).Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung.Acta Bioth. 5,11–40.Google Scholar
[10]Guttorp, P. (1991).Statistical Inference for Branching Processes.John Wiley,New York.Google Scholar
[11]Harris, T. E. (1948).Branching processes.Ann. Math. Statist. 19,474–494.Google Scholar
[12]Immel, E. R. (1951). Problems of estimation and of hypothesis testing connected with birth-and-death Markov processes. Doctoral Thesis, University of California.Google Scholar
[13]Jensen, A. (1953).Markoff chains as an aid in the study of Markoff processes.Skand. Aktuarietidskr. 36,87–91.Google Scholar
[14]Karlin, S. and McGregor, J. (1967).The number of mutant forms maintained in a population. In Proc. 5th Berkeley Symp. Math. Statist. Prob.,University of California Press,Berkeley, CA, pp. 415–438.Google Scholar
[15]Keiding, N. (1975).Maximum likelihood estimation in the birth-and-death process.Ann. Statist. 3,363–372.Google Scholar
[16]Kendall, D. G. (1948).On the generalized “birth-and-death” process.Ann. Math. Statist. 19,1–15.Google Scholar
[17]Li, Y.-F., Zio, E. and Lin, Y.-H. (2014).Methods of solutions of inhomogeneous continuous time Markov chains for degradation process modeling. In Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference, eds I. B. Frenkel et al.,John Wiley.Google Scholar
[18]Masuyama, H. (2017). Limit formulas for the normalized fundamental matrix of the northwest-corner truncation of Markov chains: matrix-infinite-product-form solutions of block-Hessenberg Markov chains. Preprint. Available at https://arxiv.org/abs/1603.07787v5.Google Scholar
[19]Melloy, B. J. and Bennett, G. K. (1993).Computing the exponential of an intensity matrix.J. Comput. Appl. Math. 46,405–413.Google Scholar
[20]Moler, C. and Van Loan, C. (1978).Nineteen dubious ways to compute the exponential of a matrix.SIAM Rev. 20,801–836.Google Scholar
[21]Moler, C. and Van Loan, C. (2003).Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later.SIAM Rev. 45,3–49.Google Scholar
[22]Pritchard, J. K., Seielstad, M. T., Perez-Lezaun, A. and Feldman, M. W. (1999).Population growth of human Y chromosomes: a study of Y chromosome microsatellites.Mol. Biol. Evol. 16,1791–1798.Google Scholar
[23]Rubin, D. B. (1984).Bayesianly justifiable and relevant frequency calculations for the applied statistician.Ann. Statist. 12,1151–1172.Google Scholar
[24]Seneta, E. (2006).Non-negative Matrices and Markov Chains,2nd edn.Springer,New York.Google Scholar
[25]Sisson, S. A., Fan, Y. and Beaumont, M. A. (eds) (2018).Handbook of Approximate Bayesian Computation.CRC Press.Google Scholar
[26]Smith, D. M. (1998).Algorithm 786: multiple-precision complex arithmetic and functions.ACM Trans. Math. Software 24,359–367.Google Scholar
[27]Van Dijk, N. M. (1992).Uniformization for nonhomogeneous Markov chains.Operat. Res. Lett. 12,283–291.Google Scholar
[28]Waugh, W. A. O’N. (1958).Conditioned Markov processes.Biometrika 45,241–249.Google Scholar
[29]Xu, J. and Minin, V. N. (2015). Efficient transition probability computation for continuous-time branching processes via compressed sensing. Preprint. Available at https://arxiv.org/abs/1503.02644v1.Google Scholar