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Asymptotic distribution of the CLSE in a critical process with immigration

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  • Rahimov, I.

Abstract

It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n2/3. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents [alpha] and [beta], respectively. We prove that if [beta] 1+2[alpha], its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When [beta]=1+2[alpha] the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.

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  • Rahimov, I., 2008. "Asymptotic distribution of the CLSE in a critical process with immigration," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1892-1908, October.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:10:p:1892-1908
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    References listed on IDEAS

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    1. Wei, C. Z. & Winnicki, J., 1989. "Some asymptotic results for the branching process with immigration," Stochastic Processes and their Applications, Elsevier, vol. 31(2), pages 261-282, April.
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    Cited by:

    1. M. González & M. Mota & I. Puerto, 2011. "Weighted conditional least square estimators for bisexual branching processes with immigration," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 607-629, November.
    2. Wagner Barreto-Souza & Sokol Ndreca & Rodrigo B. Silva & Roger W. C. Silva, 2023. "Non-linear INAR(1) processes under an alternative geometric thinning operator," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 695-725, June.

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