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Asymptotic inference for non-supercritical partially observed branching processes

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  • Rahimov, I.

Abstract

To estimate the offspring mean of a branching process one needs observed population sizes up to some generation. However, in applications very often not all individuals existing in the population are observed. Therefore the question about possibility of estimating the population mean based on partial observations is of interest. In existing literature this problem has been studied assuming that the process never becomes extinct, which is possible only in supercritical case. In the paper we consider it in subcritical and critical processes with a large number of initial ancestors. We prove that the Harris type ratio estimator remains consistent, if we have observations of a binomially distributed subsets of the population. To obtain the asymptotic normality of the estimator we modify the estimator using a “skipping” method. The proofs use the law of large numbers and the central limit theorem for random sums in the case when the number of terms and the terms in the sum are not independent.

Suggested Citation

  • Rahimov, I., 2017. "Asymptotic inference for non-supercritical partially observed branching processes," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 26-32.
    • Handle: RePEc:eee:stapro:v:126:y:2017:i:c:p:26-32
    DOI: 10.1016/j.spl.2017.02.024
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    References listed on IDEAS

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    1. Ronald Meester & Jan Koning & Mart C. M. Jong & Odo Diekmann, 2002. "Modeling and Real-Time Prediction of Classical Swine Fever Epidemics," Biometrics, The International Biometric Society, vol. 58(1), pages 178-184, March.
    2. Scott, D. J., 1978. "A central limit theorem for martingales and an application to branching processes," Stochastic Processes and their Applications, Elsevier, vol. 6(3), pages 241-252, February.
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