Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T03:07:44.971Z Has data issue: false hasContentIssue false
  • English
  • Français

On estimation of the variances for critical branching processes with immigration

Part of: Special processes Stochastic analysis Markov processes

Published online by Cambridge University Press:  14 July 2016

Chunhua Ma  and
Longmin Wang
Chunhua Ma*
Affiliation:
Nankai University
Longmin Wang*
Affiliation:
Nankai University
*
∗Postal address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China.
∗Postal address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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).

Keywords

Branching process with immigrationregular variationconditional least squares estimatorlimiting distribution

MSC classification

Secondary: 60J35: Transition functions, generators and resolvents 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H20: Stochastic integral equations 60K37: Processes in random environments
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 526 - 542
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported by the NSFC (grant numbeers 10871103 and 10971106)

References

[1] Aldous, D. (1978). Stopping times and tightness. Ann. Prob. 6, 335–340.Google Scholar
[2] Billingsley, P. (1999). Convergence of Probability Measures. John Wiley, New York.CrossRefGoogle Scholar
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[4] Dawson, D. A. and Li, Z. H. (2006). Skew convolution semigroups and affine Markov processes. Ann. Prob. 34, 1103–1142.Google Scholar
[5] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.CrossRefGoogle Scholar
[6] Hall, P. and Heyde, C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York.Google Scholar
[7] Heyde, C. C. (1974). On estimating the variance of the offspring distribution in a simple branching process. Adv. Appl. Prob. 6, 421–433.Google Scholar
[8] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes (North-Holland Math. Library 24), 2nd edn. North-Holland, Amsterdam.Google Scholar
[9] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.Google Scholar
[10] Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.Google Scholar
[11] Klimko, L. A. and Nelson, P. I. (1978). On conditional least squares estimation for stochastic processes. Ann. Statist. 6, 629–642.CrossRefGoogle Scholar
[12] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Prob. 19, 1035–1070.CrossRefGoogle Scholar
[13] Li, Z. (2005). A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Prob. 43, 289–295.Google Scholar
[14] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
[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, 275–302.Google Scholar
[16] Wei, C. Z. and Winnicki, J. (1989). Some asymptotic results for the branching process with immigration. Stochastic Process. Appl. 31, 261–282.Google Scholar
[17] Wei, C. Z. and Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Ann. Statist. 18, 1757–1773.Google Scholar
[18] Winnicki, J. (1991). Estimation of the variances in the branching process with immigration. Prob. Theory Relat. Fields 88, 77–106.CrossRefGoogle Scholar