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On estimation of the variances for critical branching processes with immigration

  • Chunhua Ma (a1) and Longmin Wang (a1)

Abstract

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

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Copyright

Corresponding author

Postal address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China.
∗∗ Email address: mach@nankai.edu.cn
∗∗∗ Email address: wanglm@nankai.edu.cn

Footnotes

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Supported by the NSFC (grant numbeers 10871103 and 10971106)

Footnotes

References

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[1] Aldous, D. (1978). Stopping times and tightness. Ann. Prob. 6, 335340.
[2] Billingsley, P. (1999). Convergence of Probability Measures. John Wiley, New York.
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
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[7] Heyde, C. C. (1974). On estimating the variance of the offspring distribution in a simple branching process. Adv. Appl. Prob. 6, 421433.
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[11] Klimko, L. A. and Nelson, P. I. (1978). On conditional least squares estimation for stochastic processes. Ann. Statist. 6, 629642.
[12] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Prob. 19, 10351070.
[13] Li, Z. (2005). A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Prob. 43, 289295.
[14] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
[15] Samorodnitsky, G., Rachev, S. T., Kurz-Kim, J. R. and Stoyanov, S. V. (2007). Asymptotic distribution of unbiased linear estimators in the presence of heavy-tailed stochastic regressors and residuals. Prob. Math. Statist. 27, 275302.
[16] Wei, C. Z. and Winnicki, J. (1989). Some asymptotic results for the branching process with immigration. Stochastic Process. Appl. 31, 261282.
[17] Wei, C. Z. and Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Ann. Statist. 18, 17571773.
[18] Winnicki, J. (1991). Estimation of the variances in the branching process with immigration. Prob. Theory Relat. Fields 88, 77106.

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On estimation of the variances for critical branching processes with immigration

  • Chunhua Ma (a1) and Longmin Wang (a1)

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