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Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

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  • Bretó, Carles
  • Ionides, Edward L.

Abstract

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-)equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Lévy-driven SDEs. We construct multivariate infinitesimally over-dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.

Suggested Citation

  • Bretó, Carles & Ionides, Edward L., 2011. "Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2571-2591, November.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:11:p:2571-2591
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    References listed on IDEAS

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    1. Christophe Andrieu & Arnaud Doucet & Roman Holenstein, 2010. "Particle Markov chain Monte Carlo methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(3), pages 269-342, June.
    2. Fan, Ruzong & Lange, Kenneth & Peña, Edsel, 1999. "Applications of a formula for the variance function of a stochastic process," Statistics & Probability Letters, Elsevier, vol. 43(2), pages 123-130, June.
    3. Varughese, M.M. & Fatti, L.P., 2008. "Incorporating environmental stochasticity within a biological population model," Theoretical Population Biology, Elsevier, vol. 74(1), pages 115-129.
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    1. Bretó, Carles, 2014. "Trajectory composition of Poisson time changes and Markov counting systems," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 91-98.
    2. Jonathan Fintzi & Jon Wakefield & Vladimir N. Minin, 2022. "A linear noise approximation for stochastic epidemic models fit to partially observed incidence counts," Biometrics, The International Biometric Society, vol. 78(4), pages 1530-1541, December.
    3. Bretó, Carles, 2012. "On the infinitesimal dispersion of multivariate Markov counting systems," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 720-725.
    4. King, Aaron A. & Lin, Qianying & Ionides, Edward L., 2022. "Markov genealogy processes," Theoretical Population Biology, Elsevier, vol. 143(C), pages 77-91.
    5. Bretó, Carles, 2012. "Time changes that result in multiple points in continuous-time Markov counting processes," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2229-2234.
    6. King, Aaron A. & Nguyen, Dao & Ionides, Edward L., 2016. "Statistical Inference for Partially Observed Markov Processes via the R Package pomp," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 69(i12).

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