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On Classes of Equivalence and Identifiability of Age-Dependent Branching Processes

  • Rui Chen (a1) and Ollivier Hyrien (a1)

Abstract

Age-dependent branching processes are increasingly used in analyses of biological data. Despite being central to most statistical procedures, the identifiability of these models has not been studied. In this paper we partition a family of age-dependent branching processes into equivalence classes over which the distribution of the population size remains identical. This result can be used to study identifiability of the offspring and lifespan distributions for parametric families of branching processes. For example, we identify classes of Markov processes that are not identifiable. We show that age-dependent processes with (nonexponential) gamma-distributed lifespans are identifiable and that Smith-Martin processes are not always identifiable.

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Copyright

Corresponding author

Postal address: Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY 14642, USA.
∗∗ Email address: rui_chen@urmc.rochester.edu
∗∗∗ Email address: ollivier_hyrien@urmc.rochester.edu

References

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[1] Gyllenberg, M. and Webb, G. F. (1990). A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol. 28, 671694.
[2] Haccou, P., Jagers, P. and Vatutin, V. A. (2007). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.
[3] Hyrien, O., Mayer-Pröschel, M., Noble, M. and Yakovlev, A. (2005). A stochastic model to analyze clonal data on multi-type cell populations. Biometrics 61, 199207.
[4] Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.
[5] Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology. Springer, New York.
[6] Redner, R. (1981). Note on the consistency of the maximum likelihood estimate for nonidentifiable distributions. Ann. Statist. 9, 225228.
[7] Sevastyanov, B. A. (1971). Branching Processes. Nauka, Moscow (in Russian).
[8] Smith, J. A. and Martin, L. (1973). Do cells cycle? Proc. Nat. Acad. Sci. 70, 12631267.
[9] Yakovlev, A. Y. and Yanev, N. M. (1989). Transient Processes in Cell Proliferation Kinetics. Springer, Berlin.

Keywords

MSC classification

On Classes of Equivalence and Identifiability of Age-Dependent Branching Processes

  • Rui Chen (a1) and Ollivier Hyrien (a1)

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