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Validity of the bootstrap in the critical process with a non-stationary immigration

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  • I. Rahimov
  • M. Omar

Abstract

In the critical branching process with a stationary immigration, the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α(n) and β(n) are finite for each n≥1 and are regularly varying sequences with non-negative exponents α and β, respectively. We prove that if α(n)→∞ and β (n)=o(nα2(n)) as n→∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least-squares estimator in the sense of convergence in probability. Monte Carlo and bootstrap simulations for the process confirm the theoretical findings in the paper and highlight the validity and utility of the bootstrap as it mimics the Monte Carlo pivots even when generation size is small.

Suggested Citation

  • I. Rahimov & M. Omar, 2011. "Validity of the bootstrap in the critical process with a non-stationary immigration," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(1), pages 1-19.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:1:p:1-19
    DOI: 10.1080/10485250903085839
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    References listed on IDEAS

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    1. Datta, Somnath & Sriram, T. N., 1995. "A modified bootstrap for branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 275-294, April.
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